SRIT / UICM007 P & S / Two Dimensional Random Variables

SRIT / UICM007 – P & S / Two Dimensional Random Variables

SRIT / M & H / M. Vijaya Kumar 1

SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY

(AN AUTONOMOUS INSTITUTION)

COIMBATORE- 641010

UICM007 & PROBABILITY AND STATISTICS

Two Dimensional Random Variables

Let be the sample space of a random experiment . Let ( ) and ( ) be

two random variables each assigning real numbers to each outcome . Then ( ) is

called two dimensional random variable.

Types of random variable

Discrete random variable

Continuous random variable

Discrete random variable Continuous random variable

1 Joint Probability

[ ] ( ) ( ) ( )

2

Marginal Probability

, - ( ) ∑ ( )

[ ] ( ) ∑ ( )

( ) ( ) ∫ ( )

( ) ( ) ∫ ( )

3

Conditional Probability

[ ⁄ ] [ ]

( )

, ⁄ - [ ]

( )

( ) ( )

( )

( ) ( )

( )

4 If and are independent

[ ] ( ) ( ) ( ) ( ) ( )



Problem: 1

( )

Find the marginal and conditional distributions.

Answer:

The marginal distributions of and are

Marginal probability

of is ( ) ( ) 1 2 3

1

( )

2

( )

Marginal probability of is

( ) ( ) ( )

( )

( )

∑ ( )

Conditional distribution of :

( ⁄ ) ( )

( )

1 2 3

1

2


( ⁄ ) ( )

( )

1 2 3

1

2



Problem: 2

The joint probability distribution function is given by ( ) ( )

Find all the marginal and conditional probability distributions. Also

find the distribution function of .

Answer:

The marginal distributions are show below

( ) ( )

0 1 2

1 ( ) ( ) ( ) ( )

2 ( ) ( ) ( ) ( )

3 ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ∑ ( )

We know that the total probability

∑ ( )

The marginal distributions of and are

Marginal probability

of is ( ) 0 1 2

1

( )

2

( )

3

( )

Marginal Probability

of is ( ) ( )

( )

( )

∑ ( )


( ⁄ ) ( )

( )



0 1 2

1

2

3

( )

( )

( )

( )

( )

( )


( ⁄ ) ( )

( )

0 1 2

1

( )

( )

2

( )

( )

3

( )

( )

Probability distribution function of

1 ( )

2 ( ) ( )

3 ( ) ( ) ( )

4 ( ) ( )

5 ( )



Problem: 3

The bivariate random variable and has the probability density function

( ) ( )

Answer:

By definition of probability density function

∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *( ( )

) (

)+

∫ *

+

∫ *

+

*

+

(

)

(

)

Problem: 4 The bivariate random variable and has the probability density function

( ) ( )( )

Answer:


∫ ∫ ( )



∫ ∫ ( )( )

∫ ( )

∫ ( )

*

+

*

+

, -

*

+ *(

) (

)+

, - [(

)

]

, - , -

Problem: 5

( ) { ( )

( ) ( ) ( ) ( ) .

/

Answer:

To find the constant C:


∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *(

) ( ( )

)+

∫ *

+

∫

*

+



*

+

( ) ∫ ( )

∫ ( )

∫ ( )

[

]

*

+

*(

) ( ( )

)+

*

+

, -

( )

By definition of conditional probaility, we have

.

/ ( )

( )

( )

( )

.

/ ( )



Problem: 6

The joint probability density function of two random variables and is given by

( ) {

i. Find the value ii. Find the mariginal density

functions of and .

Answer:

To find the constant :


∫ ∫ ( )

∫ ∫

∫

∫

, -

(

)

(

)

(

) (

)

( ) ∫ ( )

∫

*

+

(

)

( ) ( )

( ) ∫ ( )

∫



*

+

(

)

( )

Problem: 7

The joint probability density function of a two dimensional random variables is

( ) {

( ) (

)

( ) (

⁄ ) ( ) ( )

Answer:

( ) {

( ) ∫ ( )

∫ *

+

*

+

( )

( ) ∫ ( )

∫ *

+

*

+

( )



(

) ∫ ( )

∫ [

]

(

)

(

)

(

)

(

)

(

⁄ )

. /

. /

(

) ∫ ∫ ( )

∫ ∫ *

+

∫ (

)

∫ (

(

)

(

))

(

)

(

) (

)

(

⁄ )

. /

. /



( )

1

( ) ∫ ∫ (

)

∫ *

+

∫ *( )

( ) +

∫ [

( )

( ) ]

∫ [

( )

( ) ]

*

(

)

( )

( )+

[

(

)

( )]

[

(

)

( )]

( )

( )

𝑥 𝑥

𝑦

𝑦

𝑥 𝑦

𝑥 ( )

𝑦

( )



Problem: 8

The joint probability density function of two random variables & is given by

( ) { (

)

(

) (

⁄ )

Answer:

( ) ∫ ( )

∫ (

)

*

+

, -

( )

( ) ∫ ( )

∫ (

)

(

) *

+

(

) [

]

( ) (

)

(

) ( )

( ) (

)

(

)



(

)

(

)

Problem: 9

Determine whether the random variables and are independent, given their

( ) {

Also find the conditional densities of on and on .

Answer:

( ) ∫ ( )

∫ .

/

(

)

(

)

( )

( ) ∫ ( )

∫ .

/

(

)

(

)

( )



( ) ( ) (

) (

)

( ) ( ) ( )

and are not independent.

(

) ( )

( )

(

)

.

/ ( )

( )

Problem: 10

If the joint PDF of the function and is given by

( ) {

( )

( ) ( ) ( ) ( ) ( ) ( )

Answer:

( ) ∫ ∫

( )



∫ *

+

∫ *(

) (

)+

∫ [

]

∫ [

]

*

+

[

]

( )

( ) ( )

( )

( ) ∫ ∫

( )

∫ *

+

∫ *(

) (

)+

∫ [

]

∫ [

]

*

+

*

+

[

]



( )

( ) ( )

( )

( )

( )

∫ ∫ ( )

∫ *

+

∫ * ( )

( )

( )+

∫ * ( )

( )

( )+

* ( )

( )

( )

( ) (

)+

* (

) (

(

))+

[

]

( )

( )

( )

( )

𝑥

𝑥

𝑦

𝑦

𝑥 𝑦

𝑥

𝑦

( )

𝑦



Problem: 11

Let and have the joint p.d.f given by

( ) {

( )

( ) [(

) (

)] ( ) [(

) (

)]

( ) ( ) ( ) ( )

Answer:

( ) {

( ) ∫ ( )

∫

*

+

, -

( ) ( )

( ) ∫ ( )

∫

(

)

( )

( )

[(

) (

)] ∫ ∫

∫

∫

, -



*

+

*

+

*(

)

+

*(

)

+

[

] [

]

[(

) (

)]

[(

)] 0.

/ .

/1

.

/

[(

) (

)] ∫ ∫

∫

∫

, -

*

+

*

+

*(

)

+

*(

)

(

)

+

[

] [

]

[

]

[(

) (

)]

(

) ∫ ( )

∫

*

+



*(

)

(

)

+

(

)

(

) . /

. /

( ) ( ) ( ) ( )

and

( ) ∫ ( )

∫ ( )

∫ ( )

*

+

[

]

( )

( ) ∫ ( )

∫

∫

*

+

, -

( )



Correlation

If the change in one variable affects the change in the other variable, then the

variable are said to be correlated.

Positive correlation:

If the two variable deviate in the same direction (i.e., increase or decrease) in one

variable in a corresponding (increase or decrease) in other variable is said to be positive

correlation.

Ex: Income and expenditure

Negative correlation:

If the two variable deviate in opposite direction (i.e., increase or decrease) in one

variable in a corresponding (decrease or increase) in other variable is said to be negative

correlation.

Ex: Price and demand of a product

Rank correlation

Sometimes there doesn’t exist a marked linear relationship between two random

variables but a monotonic relation (if one increases, the other also increases or instead,

decreases) is clearly noticed. Pearson’s Correlation Coefficient evaluation, in this case,

would give us the strength and direction of the linear association only between the

variables of interest. Herein comes the advantage of the Spearman Rank Correlation

methods, which will instead, give us the strength and direction of the monotonic relation

between the connected variables. This can be a good starting point for further evaluation.

The Spearman Rank-Order Correlation Coefficient

The Spearman’s Correlation Coefficient, represented by or by , is a

nonparametric measure of the strength and direction of the association that exists between

two ranked variables. It determines the degree to which a relationship is monotonic, i.e.,

whether there is a monotonic component of the association between two continuous or

ordered variables.

Monotonicity is “less restrictive” than that of a linear relationship. Although

monotonicity is not actually a requirement of Spearman’s correlation, it will not be

https://www.toppr.com/guides/business-economics-cs/mathematics-of-finance-and-elementary-probability/random-variables/

https://www.toppr.com/guides/business-economics-cs/mathematics-of-finance-and-elementary-probability/random-variables/

https://www.toppr.com/guides/business-mathematics-and-statistics/correlation-and-regression/karl-pearsons-coefficient-correlation/

https://www.toppr.com/guides/business-studies/directing/elements-of-directing/

https://www.toppr.com/guides/business-studies/forms-of-business-organisations/introduction-and-evaluation-to-forms-of-business-organisations/

https://www.toppr.com/guides/business-mathematics-and-statistics/correlation-and-regression/

https://www.toppr.com/guides/maths/linear-programming/linear-programming-problem/



meaningful to pursue Spearman’s correlation to determine the strength and direction of a

monotonic relationship if we already know the relationship between the two variables is

not monotonic.

Spearman Ranking of the Data

We must rank the data under consideration before proceeding with the Spearman’s

Rank Correlation evaluation. This is necessary because we need to compare whether on

increasing one variable, the other follows a monotonic relation (increases or decreases

regularly) with respect to it or not.

Thus, at every level, we need to compare the values of the two variables. The method of

ranking assigns such ‘levels’ to each value in the dataset so that we can easily compare it.

Assign number 1 to (the number of data points) corresponding to the variable

values in the order highest to lowest.

In the case of two or more values being identical, assign to them the arithmetic mean

of the ranks that they would have otherwise occupied.

For example, Selling Price values given: 28.2, 32.8, 19.4, 22.5, 20.0, 22.5 The

corresponding ranks are: 2, 1, 5, 3.5, 4, 3.5 The highest value 32.8 is given rank 1, 28.2 is

given rank 2,…. Two values are identical (22.5) and in this case, the arithmetic means of

ranks that they would have otherwise occupied (3+42) has to be taken.

Spearman’s Rank Correlation formula

∑

( )

where is the number of data points of the two variables and is the difference in the

ranks of the ith element of each random variable considered. The Spearman correlation

coefficient, , can take values from to .

https://www.toppr.com/guides/reasoning-ability/number-ranking-time-sequence/ranking-sequence/

https://www.toppr.com/guides/maths/data-handling/arithmetic-mean/

https://www.toppr.com/guides/quantitative-aptitude/profit-and-loss/selling-price/

https://www.toppr.com/guides/maths/sequences-and-series/means/



I. If the value of is indicates a perfect association of ranks

II. If the value of is indicates no association between ranks and

III. If the value of is indicates a perfect negative association of ranks.

IV. If the value of near to zero, the weaker the association between the ranks.

Merits of Rank Correlation Coefficient

1. Spearman’s rank correlation coefficient can be interpreted in the same way as the Karl

Pearson’s correlation coefficient;

2. It is easy to understand and easy to calculate;

3. If we want to see the association between qualitative characteristics, rank correlation

coefficient is the only formula;

4. Rank correlation coefficient is the non-parametric version of the Karl Pearson’s product

moment correlation coefficient; and

5. It does not require the assumption of the normality of the population from which the

sample observations are taken.

Demerits of Rank Correlation Coefficient

1. Product moment correlation coefficient can be calculated for bivariate frequency

distribution but rank correlation coefficient cannot be calculated; and

2. If , this formula is time consuming.

Problem: 12

The following table provides data about the percentage of students who have free

university meals and their CGPA scores. Calculate the Spearman’s Rank Correlation

between the two and interpret the result.

State University % of students having free meals % of students scoring above 8.5 CGPA

Pune 14.4 54

Chennai 7.2 64

Delhi 27.5 44

Kanpur 33.8 32

Ahmedabad 38.0 37

https://www.toppr.com/guides/quantitative-aptitude/percentages/inverse-case-value-from-percentage/



Indore 15.9 68

Guwahati 4.9 62

Answer:

Let us first assign the random variables to the required data –

X – % of students having free meals

Y – % of students scoring above 8.5 CGPA

Before proceeding with the calculation, we’ll need to assign ranks to the data

corresponding to each state university. We construct the table for the rank as below

Rank in X Rank in Y

14.4

7.2

27.5

33.8

38.0

15.9

4.9

∑ ∑

Rank Correlation

∑

( )

(

( )) , -

(

)

The result shows that a strong negative coefficient of correlation. That is the highest

percentage of students consuming free meals tend to have the least successful results.


https://www.toppr.com/guides/economics/correlation/correlation-in-a-whole/



Problem: 13

Compute the coefficient of rank correlation between sales and advertisement expressed in

thousands of dollars from the following data:

Sales 90 85 68 75 82 80 95 70

Advertisement 7 6 2 3 4 5 8 1

Answer:

Rank in X Rank in Y

90 2 7 2 0

85 3 6 3 0

68 8 2 7 1

75 6 3 6 0

82 4 4 5 1

80 5 5 4 1

95 1 8 1 0

70 7 1 8 1

∑ ∑

Rank Correlation

∑

( )

(

( )) , -

(

)

The result shows that a strong positive coefficient of correlation. Hence there is a very

good amount of agreement between sales and advertisement.


https://www.toppr.com/guides/economics/correlation/correlation-in-a-whole/



Problem: 14

Find the rank correlation co efficient from the following data.

Rank in X 1 2 3 4 5 6 7

Rank in Y 4 3 1 2 6 5 7

Answer:

∑ ∑

Rank Correlation

∑

( )

(

( )) , -

(

)

Problem: 15

The ranks of some 16 students in mathematics and physics are as follows. Find the

rank correlation for the proficiency in mathematics and physics.

Rank in Math’s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Rank in Physics 1 10 3 4 5 7 2 6 8 11 15 9 14 12 16 13



Answer:

∑ ∑

Rank Correlation

∑

( )

(

( )) , -

(

)



Problem: 16

Suppose we have ranks of 5 students in three subjects Computer, Physics and Statistics and

we want to test which two subjects have the same trend.

Rank in Computer 2 4 5 1 3

Rank in Physics 5 1 2 3 4

Rank in Statistics 2 3 5 4 1

Answer:

In this problem ranks are directly given.

Rank

in X

Rank

in Y

Rank

in Z

2 5 2

4 1 3

5 2 5

1 3 4

3 4 1

∑ ∑

∑

Rank Correlation

∑

( )

I. Rank correlation between computer and physics:

∑

( )

(

( )) , -

(

)

II. Rank correlation between physics and statistics:

∑

( )

(

( )) , -



(

)

III. Rank correlation between computer and statistics:

∑

( )

(

( )) , -

(

)

Since and are negative which indicates that Computer and Physics also Physics and

Statistics have opposite trends. But indicates that Computer and Statistics have same

trend.

Repeated Ranks

If the value is repeated in any row in any series or series in times, then we

have add the correction factor in the rank correlation formula

( )

(∑

( ))

Problem: 17

Determine the rank correlation co efficient for the following data.

68 64 75 50 64 80 75 40 55 64

62 58 68 45 81 60 68 48 50 70

Answer:

Rank in Rank in



∑ ∑

To find Correction Factor:

In series value 75 is repeated two times

( )

In series value 64 is repeated three times

( )


( )

Rank Correlation:

(∑ ( )

)

(

( )) , -

(

)



Problem: 18

The sample of 12 fathers and their eldest sons have the following data about their

heights in inches.

Fathers 65 63 67 64 68 62 70 66 68 67 69 71

Sons 68 66 68 65 69 66 68 65 71 67 68 70

Answer:

Rank in Rank in

∑ ∑

To find Correction Factor:


( )


( )

In series value 68 is repeated four times

( )




( )


( )

Rank Correlation:

(∑

( ))

(

( )) , -

( )

Karl – pearson’s co efficient of Correlation

Karl Pearson’s Coefficient of Correlation is widely used mathematical method

wherein the numerical expression is used to calculate the degree and direction of the

relationship between linear related variables.

Pearson’s method, popularly known as a Pearsonian Coefficient of Correlation, is the

most extensively used quantitative methods in practice. The coefficient of correlation is

denoted by .

If the relationship between two variables and is to be ascertained, then the

following formula is used:

( ) ( ) ( )

( ) ( ) ( ) ( )

∑

∑

∑

( ) ( ) , ( )- ( )

( ) , ( )-

∑

∑



Note:

If ( ) then ( ) ( ) ( )

If ( ) ( ) then and are uncorrelated.

Problem: 19

Find the covariance of and if the random variable ( ) having joint probability

density function ( ) {

Answer:

( ) ∫ ( )

∫ ( )

*

+

( ) (

)

( ) (

)

( ) ∫ ( )

∫ (

)

∫ .

/

*

+

[

]

( )

( )



( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *

+

*

+

( )

( ) ( ) ( ) ( )

Problem: 20

The joint probability density function of a random variable ( ) is

( ) and Find the covariance of and .

Answer:

( ) ∫ ( )

∫

*

+



*

+ , -

( )

( ) ∫ ( )

∫

(

)

(

)

( )

( ) ∫ ( )

∫

, -

( )

( )

( ) ∫ ( )

∫

(

( ) )

( *

+)

( )

( ) ∫ ∫ ( )



∫ ∫

∫ (

( ) )

, -

∫ ( *

+)

∫

(

)

(

)

( )

( ) ( ) ( ) ( )

Since Covariance of and is 0. Therefore and are independent.

Problem: 21

Two random variables and having the joint probability density function given by

( ) { ( )

i. Find the value ii. Find the marginal

density function of and iii. Find the correlation co efficient of and .

Answer:

I. To find the value :

By definition of total probability density function

∫ ∫ ( )

∫ ∫ ( )

∫ (

)



∫ (

)

(

)

(

)

( ) ∫ ( )

∫ ( )

*

+

( )

(

)

( ) ∫ ( )

∫ ( )

*

+

( )

.

/

( ) ∫ ( )

∫

(

)

∫ (

)



(

)

(

)

( )

( ) ∫ ( )

∫

(

)

∫ (

)

(

)

(

)

( )

( ) ∫ ( )

∫

.

/

∫ (

)

(

)

(

)

( )

( ) ∫ ( )

∫

.

/



∫ (

)

(

)

(

)

( )

( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ (

)

∫ (

)

(

)

(

)

( )

√ ( ) * ( )+

√

(

)

√ ( ) * ( )+

√

(

)

( ) ( ) ( ) ( )

(

)



( ) ( )

( )

Problem: 22

The joint probability density function of a two dimensional random variables are

( ) ( ) ( )

Answer:

( ) ( )

1

( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *( )

( ) +

∫ *( )

( ) +

∫ *( )

( )+

( )

𝑥

𝑥

𝑦

𝑦

𝑥 𝑦

𝑥 ( )

𝑦

( )



*( )

( )

(

)+

[

(

) (

)]

(

)

( )

( )

( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *( )

( ) +

∫ *( )

( ) +

∫ *( )

( )+

*( )

( )

(

)+

[

(

) (

)]

(

)

( )

( )



( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *( )

( ) +

∫ *( )

( ) +

∫ [

( )

( )]

*

(

)

(

)+

[

(

)

(

)]

(

)

( )

√ ( ) * ( )+

√

(

)

( ) ( ) ( ) ( )



( ) ( )

( )

Problem: 23

Let and be random variables having joint probability density function

( ) {

( )

Answer:

( ) ∫ ( )

∫ ( )

*

+

[

]

( )

( )

( ) ∫ ( )

∫ (

)

∫ ( )



*

+

[

]

( )

( )

( ) ∫ ( )

∫ (

)

∫ ( )

*

+

[

]

( )

( )

( ) ∫ ∫ ( )

∫ ∫

( )

∫ ∫ ( )

∫ *

+



∫ [

]

*

+

[

]

( )

√ ( ) * ( )+

√

(

)

( ) ( ) ( ) ( )

( ) ( )

( )

Problem: 24


( ) { ( )

Answer:

To find the constant C:


∫ ∫ ( )

∫ ∫ ( )



∫ *

+

∫ (

)

*

+

*

+

( ) ∫ ( )

∫ ( )

*

+

*

+

, -

, -

( )

( )

( ) ∫ ( )

∫ (

)

∫ ( )



*

+

[

]

( )

( )

( ) ∫ ( )

∫ (

)

∫ ( )

*

+

*

+

[

]

( )

( )

( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ *

+

∫ *

+



*

+

*

+

[

]

( )

√ ( ) * ( )+

√ (

)

( ) ( ) ( ) ( )

( ) ( )

( )

Problem: 25


( ) { ,( ) ( )- ( )

Show that and are

uncorrelated but not independent.

Answer:

( ) { , -

To find the constant :




∫ ∫ ( )

∫ ∫ , -

∫ *

+

∫ [

]

*

+

[

]

(

)

( ) ∫ ( )

∫ , -

*

+

[

]

( ) [

]

( ) [

]

( ) ∫ ( )

∫ [

]

∫ (

)



(

)

(

)

( )

( )

( ) ∫ ∫ ( )

∫ ∫ ( )

∫ ∫ ( )

∫ (

)

∫ (

)

(

)

(

)

(

)

(

)

( )

( ) ( ) ( ) ( )

and are uncorrelated.



( ) ( ) [

] [

] ( )

and are not independent.

Problem: 26

The joint pdf of and is ( ) { ( )

Find covariance of

and . Are and are independent?

Answer:

( ) {

( ) ∫ ( )

∫

[

]

(

)

(

) , -

( )

( ) ∫ ( )

∫

[

]

(

)

(

) , -

( )



( ) ∫ ( )

∫

(

( ) )

, -

,( ) ( )- , -

( )

( )

( ) ∫ ∫ ( )

∫ ∫

∫

∫

, -

(

( ) )

(

( ) )

,( ) ( )- ,( ) ( )-

( )

( ) ( ) ( ) ( )

and are uncorrelated.

( ) ( ) ( )

and are independent.

Problem: 27

Find the correlation co efficient for the following data

X 10 14 18 22 26 30

Y 18 12 24 6 30 36

Answer:



10 18 180 100 324

14 12 168 196 144

18 24 432 324 576

22 6 132 484 36

26 30 780 676 900

30 36 1080 900 1296

∑ ∑ ∑ ∑ ∑

∑

∑

√∑

( )

√

( )

√∑

( )

√

( )

( ) ∑

( ) ( )

Problem: 28

The table below shows the number of absences, , in a Calculus course and the final exam

grade, , for 7 students. Find the correlation coefficient.

X 1 0 2 6 4 3 3 Y 95 90 90 55 70 80 85

Answer:



1 95 1 9025 95

0 90 0 8100 0

2 90 4 8100 180

6 55 36 3025 330

4 70 16 4900 280

3 80 9 6400 240

3 85 9 7225 255

∑ ∑ ∑ ∑ ∑

∑

∑

√∑

( ) √

√∑

( ) √

( ) ∑

( ) ( )

Problem: 29

The marks obtained by 10 students in Mathematics and Statistics are given below.

Find the correlation coefficient between the two subjects.

Marks in math’s 75 30 60 80 53 35 15 40 38 48

Marks in Stats 85 45 54 91 58 63 35 43 45 44

Answer:



75 85 6375 5625 7225

30 45 1350 900 2025

60 54 3240 3600 2916

80 91 7280 6400 8281

53 58 3074 2809 3364

35 63 2205 1225 3969

15 35 525 225 1225

40 43 1720 1600 1849

38 45 1710 1444 2025

48 44 2112 2304 1936

∑ ∑ ∑ ∑ ∑

∑

∑

√∑

( ) √

√∑

( ) √

( ) ∑

( ) ( )

Problem: 30

Compute the coefficient of correlation between X and Y using the following data:



X 1 3 5 7 8 10

Y 8 12 15 17 18 20

Answer:

1 8 8 1 64

3 12 36 9 144

5 15 75 25 225

7 17 119 49 289

8 18 144 64 324

10 20 200 100 400

∑ ∑ ∑ ∑ ∑

∑

∑

√∑

( )

√

√∑

( )

√

( ) ∑

( ) ( )



Problem: 31

Calculate the coefficient of correlation for the following data:

9 8 7 6 5 4 3 2 1

15 16 14 13 11 12 10 8 9

Answer:

9 15 135 81 225

8 16 128 64 256

7 14 98 49 196

6 13 78 36 169

5 11 55 25 121

4 12 48 16 144

3 10 30 9 100

2 8 16 4 64

1 9 9 1 81

∑ ∑ ∑ ∑ ∑

∑

∑

√∑

( )

√

√∑

( )

√

( ) ∑



( ) ( )

Regression

Regression analysis is most often used for prediction. The goal in regression analysis

is to create a mathematical model that can be used to predict the values of a dependent

variable based upon the values of an independent variable. In other words, we use the

model to predict the value of when we know the value of . (The dependent variable is

the one to be predicted). Correlation analysis is often used with regression analysis

because correlation analysis is used to measure the strength of association between the

two variables and .

In regression analysis involving one independent variable and one dependent

variable the values are frequently plotted in two dimensions as a scatter plot. The scatter

plot allows us to visually inspect the data prior to running a regression analysis. Often this

step allows us to see if the relationship between the two variables is increasing or

decreasing and gives only a rough idea of the relationship. The simplest relationship

between two variables is a straight-line or linear relationship. Of course the data may well

be curvilinear and in that case we would have to use a different model to describe the

relationship (we will deal only with linear relationship’s for now). Simple linear regression

analysis finds the straight line that best fits the data

Definition:

Regression is mathematical measure of the average relationship between two or

more variables in terms of original limits of the data.

The equation of line of regression of on is

( )



The equation of line of regression of on is

( )

Regression co efficient

Correlation co efficient √

Problem: 32

Obtain the equations of the lines of regression from the following data:

1 2 3 4 5 6 7

9 8 10 12 11 13 14

Answer:

1 9 9 1 81

2 8 16 4 64

3 10 30 9 100

4 12 48 16 144

5 11 55 25 121

6 13 78 36 169

7 14 98 49 196

Total 334 140 875

∑

∑

√∑

( ) √

√∑

( ) √



( ) ( ) ( ) ( )

∑

Correlation coefficient

( )

The line of regression of on is

( )

( )


( )

( )

Problem: 33

From the following data find

The two regression lines

The co efficient of correlation between the marks in economics and statistics

The most likely marks in statistics when marks in economics are 30.

Marks in Economics: 25 28 35 32 31 36 29 38 34 32

Marks in Statistics: 43 46 49 41 36 32 31 30 33 39

Answer:

25 43 1075 625 1849

28 46 1288 784 2116

35 49 1715 1225 2401

32 41 1312 1024 1681

31 36 1116 961 1296

36 32 1152 1296 1024

29 31 899 841 961

38 30 1140 1444 900



34 33 1122 1156 1089

32 39 1248 1024 1521

Total 12067 10380 14838

∑

∑

√∑

( )

√

√∑

( )

√

( ) ( ) ( ) ( )

∑


( )


( )

( )

( )


( )

( )

( )


( )

Problem: 34



A tyre manufacturing company is interested in removing pollutants from the exhaust at the

factory, and cost is a concern. The company has collected data from other companies

concerning the amount of money spent on environmental measures and the resulting

amount of dangerous pollutants released (as a percentage of total emissions)

Money

spent

(Rupees in

lakhs

8.4 10.2 16.5 21.7 9.4 8.3 11.5 18.4 16.7 19.3 28.4 4.7 12.3

Percentage

of

dangerous

pollutants

35.9 31.8 24.7 25.2 36.8 35.8 33.4 25.4 31.4 27.4 15.8 31.5 28.9

a) Compute the regression equation.

b) Predict the percentage of dangerous pollutants released when Rs. 20,000 is spent on

control measures.

Answer:

S. No

1 8.4 35.9 70.56 1288.8 301.56

2 10.2 31.8 104.04 1011.2 324.36

3 16.5 24.7 272.25 610.09 407.55

4 21.7 25.2 470.89 635.04 546.84

5 9.4 36.8 88.36 1354.2 345.92

6 8.3 35.8 68.89 1281.6 297.14

7 11.5 33.4 132.25 1115.6 384.1

8 18.4 25.4 338.56 645.16 467.36

9 16.7 31.4 278.89 985.96 524.38

10 19.3 27.4 372.49 750.76 528.82

11 28.4 15.8 806.56 249.64 448.72

12 4.7 31.5 22.09 992.25 148.05

13 12.3 28.9 151.29 835.21 355.47



Total ∑ 185.8 ∑ 384 ∑ 3177 ∑ 11756 ∑ 5080

a) Regression equation

∑

∑

√∑

( )

√

√∑

( )

√

( ) ( ) ( ) ( )

∑


( )


( )

( )

( )


( )

( )

( )

b) When Rs. 20,000 ( ) is spent on control then the percentage of

dangerous pollutants released is

( )

Problem: 35

The quantity of a raw material purchased by a company at the specified prices during the

12 months of 1992 is given



MONTH PRICE/KG QUANTITY (KG)

Jan 96 250

Feb 110 200

Mar 100 250

Aprl 90 280

May 86 300

June 92 300

July 112 220

Aug 112 220

Sep 108 200

Oct 116 210

Nov 86 300

Dec 92 250

Find the regression equation based on the above data

Can you estimate the appropriate quantity likely to be purchased if the price shoot

upon Rs 124/kg?

Hence or otherwise obtain the coefficient of correlation between the price prevailing

and the quantity demanded

Answer:

S. No

1 96 250 9216 62500 24000

2 110 200 12100 40000 22000

3 100 250 10000 62500 25000

4 90 280 8100 78400 25200

5 86 300 7396 90000 25800

6 92 300 8464 90000 27600

7 112 220 12544 48400 24640

8 112 220 12544 48400 24640

9 108 200 11664 40000 21600

10 116 210 13456 44100 24360

11 86 300 7396 90000 25800



12 92 250 8464 62500 23000

Total ∑ 1200 ∑ 2980 ∑ 121344 ∑ 756800 ∑ 293640

a) Regression equation

∑

∑

√∑

( )

√

√∑

( )

√

( ) ( ) ( ) ( )

∑


( )


( )

( )

( )


( )

( )

( )

b) Estimation of the appropriate quantity likely to be purchased if the price shoot upon

Rs 124/kg,

Given price short , then

( )

c) Correlation coefficient



Problem: 36

The two lines of regression are . The

variance of is 9. Find (i) The mean values of and (ii) correlation coefficient between

and .

Answer:

Since both the lines of regression passes through the mean values and .

The point ( ) must satisfy the two lines.

( )

( )

Solving ( ) and ( ), we get

The mean values of and are

Consider the line is a regression line on .



√

√



Correlation co efficient

Problem: 37

The two lines of regression are . Find ( ) and

correlation coefficient between and .

Answer:



( )

( )





√

√



Correlation co efficient

Problem: 38

The regression equation of and is . If the mean value of

and the correlation coefficient.

Answer:

Given the regression equation of and is

.

Since the line of regression passes through ( ), then

Also given mean value of is

( )

Hence mean value of is 48.

which is the line of regression of on .

[

]



( )

( )

. /

Problem: 39

If and are uncorrelated random variables with variances and . Find the


Answer:

Given that ( ) ( )

Let us take and

Given that both and are uncorrelated.

Now ( ) ( )

( ) ( ) , ( ) ( ) ( )-

( )

and ( ) ( )

( ) ( ) , ( ) ( ) ( )-

( )

( ) ,( )( )-

( )



( ) ( )

( ) ,( )- ( ) ( )

( ) ,( )- ( ) ( )

Now ( ) ( ) ( ) ( )

( ) ( ) * ( ) ( )+ * ( ) ( )+

( ) ( ) ,* ( )+ * ( )+ -

, ( ) * ( )+ - , ( ) * ( )+ -

( ) ( )

( ) ( )

Problem: 40

If the independent random variables and have the variances 3 and

respectively, find the correlation coefficient between and .

Answer:

Given that ( ) ( )

Let us take and

Given that both and are uncorrelated.

Now ( ) ( )

( ) ( ) , ( ) ( ) ( )-

( )

√

and ( ) ( )

( ) ( ) , ( ) ( ) ( )-

( )

√

( ) ,( )( )-



( )

( ) ( )

( ) ,( )- ( ) ( )

( ) ,( )- ( ) ( )

Now ( ) ( ) ( ) ( )

( ) ( ) * ( ) ( )+ * ( ) ( )+

( ) ( ) ,* ( )+ * ( )+ -

, ( ) * ( )+ - , ( ) * ( )+ -

( ) ( )

( ) ( )

√ √

Two Marks

1. Define the joint pmf of a two-dimensional discrete random variable.

Answer:

The function ( ) ( ) is called the joint pmf of a two-

dimensional discrete random variable and and and is denoted by .

2. The joint pmf of two random variables X and Y is given by

( ) {

Determine the value of the constant k .

Answer:


1 2 3

1 k 2k 3k

2 2k 4k 6k

3 3k 6k 9k


∑ ( )



3. Let X and Y be two discrete random variables with joint probability mass

( ) {

( )

probability mass functions of X and Y .

Answer:


( ) ( )

1 2

1 3/18 5/18 ( )

2 4/18 6/18 ( )

( ) ( ) ( ) ( )

4. The joint probability mass function of a two dimensional random variable ( ) is

given by ( ) ( ) , where is constant. Find the value .

Answer:

( ) ( )

1 2

1 3k 5k ( )

2 4k 6k ( )

∑ ( )


∑ ( )

5. Find the value of , if ( ) ( )( ) and ( ) ,

otherwise, is to be the joint density function.

Answer:

By definition of total probability density function



∫ ∫ ( )

∫ ∫ ( )( )

∫ ( )

∫ ( )

*

+

*

+

, -

[(

) ] [(

) ]

[

] [

]

6. Let and be continuous random variables with joint probability density function

( ) ( )

( ) ⁄ .

/

Answer:

.

/ ( )

( )

( ) ( ) ∫ ( )

∫ ( )

*

+

*(

) ( ( )

)+

*

+

, -

( )



.

/ ( )

( )

( )

( )

.

/

( ) ( )

. Find ( ).

Answer:

( )

1

0 1

( ) ∫ ∫ (

)

∫ *

+

∫ *

+

∫ [

]

*

+

( )

( )

𝑥 𝑥

𝑦

𝑦

𝑥 𝑦

𝑥 ( )

𝑦

( )



8. Find the marginal density functions of and if

( ) {

( )

Answer:

( ) ∫ ( )

∫ ( )

(

)

(

)

( ) ∫ ( )

∫ ( )

(

)

(

)

9. Find the acute angle between the two lines of regression, assuming the two

lines of regression.

Answer:

(

)

10. If the joint pdf of ( ) is ( ) { ( )

Check whether

and are independent? Answer:

( ) ∫ ( )



∫

[

]

(

)

(

)

( )

( )

( ) ( )

( )

( )

and are independent.

11. The joint pdf of the random variable ( ) is given by ( ) ( )

. Find the value of .

Answer:


∫ ∫ ( )

∫ ∫ ( )

∫

∫

, -

Take ; When and

∫

∫

(

)

(

)

(

) (

)

(

) (

)

12. The two regression equations of two random variables and are



and . Find the mean values of and .

Answer:



( )

( )


Questions Problem: 1

( )

Find the marginal and conditional distributions.

Problem: 2

The joint probability distribution function is given by ( ) ( )

Find all the marginal and conditional probability distributions. Also

find the distribution function of .

Problem: 3

The bivariate random variable and has the probability density function

( ) ( )

Problem: 4 The bivariate random variable and has the probability density function

( ) ( )( )

Problem: 5

( ) { ( )

( ) ( ) ( ) ( ) .

/

Problem: 6

The joint probability density function of two random variables and is given by



( ) {

i. Find the value ii. Find the mariginal density

functions of and .

Problem: 7

The joint probability density function of a two dimensional random variables is

( ) {

( ) (

)

( ) (

⁄ ) ( ) ( )

Problem: 8

The joint probability density function of two random variables & is given by

( ) { (

)

(

) (

⁄ )

Problem: 9

Determine whether the random variables and are independent, given their

( ) ,

Also find the conditional densities of on and on .

Problem: 10

If the joint PDF of the function and is given by

( ) {

( )

( ) ( ) ( ) ( ) ( ) ( )

Problem: 11

Let and have the joint p.d.f given by

( ) {

( )

( ) [(

) (

)] ( ) [(

) (

)]

( ) ( ) ( ) ( )



Problem: 12

The following table provides data about the percentage of students who have free

university meals and their CGPA scores. Calculate the Spearman’s Rank Correlation

between the two and interpret the result.

State University % of students having free meals % of students scoring above 8.5 CGPA

Pune 14.4 54 Chennai 7.2 64

Delhi 27.5 44 Kanpur 33.8 32

Ahmedabad 38.0 37 Indore 15.9 68

Guwahati 4.9 62 Problem: 13

Compute the coefficient of rank correlation between sales and advertisement expressed in

thousands of dollars from the following data:

Sales 90 85 68 75 82 80 95 70

Advertisement 7 6 2 3 4 5 8 1

Problem: 14

Find the rank correlation co efficient from the following data.

Rank in X 1 2 3 4 5 6 7

Rank in Y 4 3 1 2 6 5 7

Problem: 15

The ranks of some 16 students in mathematics and physics are as follows. Find the

rank correlation for the proficiency in mathematics and physics.

Rank in Math’s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Rank in Physics 1 10 3 4 5 7 2 6 8 11 15 9 14 12 16 13

Problem: 16

Suppose we have ranks of 5 students in three subjects Computer, Physics and Statistics and

we want to test which two subjects have the same trend.

Rank in Computer 2 4 5 1 3

Rank in Physics 5 1 2 3 4

Rank in Statistics 2 3 5 4 1

Problem: 17

https://www.toppr.com/guides/quantitative-aptitude/percentages/inverse-case-value-from-percentage/



Determine the rank correlation co efficient for the following data.

68 64 75 50 64 80 75 40 55 64

62 58 68 45 81 60 68 48 50 70

Problem: 18

The sample of 12 fathers and their eldest sons have the following data about their

heights in inches.

Fathers 65 63 67 64 68 62 70 66 68 67 69 71

Sons 68 66 68 65 69 66 68 65 71 67 68 70

Problem: 19

Find the covariance of and if the random variable ( ) having joint probability

density function ( ) {

Problem: 20

The joint probability density function of a random variable ( ) is

( ) and Find the covariance of and .

Problem: 21

Two random variables and having the joint probability density function given by

( ) { ( )

i. Find the value ii. Find the marginal

density function of and iii. Find the correlation co efficient of and .

Problem: 22

The joint probability density function of a two dimensional random variables are

( ) ( ) ( )

Problem: 23


( ) {

( )

Problem: 24


( ) { ( )

Problem: 25




( ) { ,( ) ( )- ( )

Show that and are

uncorrelated but not independent. Problem: 26

The joint pdf of and is ( ) { ( )

Find covariance of

and . Are and are independent?

Problem: 27

Find the correlation co efficient for the following data

X 10 14 18 22 26 30

Y 18 12 24 6 30 36

Problem: 28

The table below shows the number of absences, , in a Calculus course and the final exam

grade, , for 7 students. Find the correlation coefficient.

X 1 0 2 6 4 3 3

Y 95 90 90 55 70 80 85

Problem: 29

The marks obtained by 10 students in Mathematics and Statistics are given below.

Find the correlation coefficient between the two subjects.

Marks in math’s 75 30 60 80 53 35 15 40 38 48

Marks in Stats 85 45 54 91 58 63 35 43 45 44

Problem: 30

Compute the coefficient of correlation between X and Y using the following data:

X 1 3 5 7 8 10

Y 8 12 15 17 18 20

Problem: 31

Calculate the coefficient of correlation for the following data:

9 8 7 6 5 4 3 2 1

15 16 14 13 11 12 10 8 9



Problem: 32

Obtain the equations of the lines of regression from the following data:

1 2 3 4 5 6 7

9 8 10 12 11 13 14

Problem: 33

From the following data find

The two regression lines

The co efficient of correlation between the marks in economics and statistics


Marks in Economics: 25 28 35 32 31 36 29 38 34 32

Marks in Statistics: 43 46 49 41 36 32 31 30 33 39

Problem: 34

A tyre manufacturing company is interested in removing pollutants from the exhaust at the

factory, and cost is a concern. The company has collected data from other companies

concerning the amount of money spent on environmental measures and the resulting

amount of dangerous pollutants released (as a percentage of total emissions)

Money spent

(Rupees in lakhs

8.4 10.2 16.5 21.7 9.4 8.3 11.5 18.4 16.7 19.3 28.4 4.7 12.3

Percentage of

dangerous pollutants

35.9 31.8 24.7 25.2 36.8 35.8 33.4 25.4 31.4 27.4 15.8 31.5 28.9

a) Compute the regression equation.

b) Predict the percentage of dangerous pollutants released when Rs. 20,000 is spent on

control measures.

Problem: 35

The quantity of a raw material purchased by a company at the specified prices during the

12 months of 1992 is given

MONTH PRICE/KG QUANTITY (KG) Jan 96 250 Feb 110 200



Mar 100 250 Aprl 90 280 May 86 300 June 92 300 July 112 220 Aug 112 220 Sep 108 200 Oct 116 210 Nov 86 300 Dec 92 250

Find the regression equation based on the above data

Can you estimate the appropriate quantity likely to be purchased if the price shoot

upon Rs 124/kg?

Hence or otherwise obtain the coefficient of correlation between the price prevailing

and the quantity demanded

Problem: 36

The two lines of regression are . The

variance of is 9. Find (i) The mean values of and (ii) correlation coefficient between

and .

Problem: 37

The two lines of regression are . Find ( ) and


Problem: 38

The regression equation of and is . If the mean value of

and the correlation coefficient.

Problem: 39

If and are uncorrelated random variables with variances and . Find the


Problem: 40

If the independent random variables and have the variances 3 and

respectively, find the correlation coefficient between and .



“The best brains of the nation may be found on

the last benches of the classroom.”

― Dr. A. P. J. Abdul Kalam

Documents

SRIT / UICM007 P & S / Two Dimensional Random Variables