RV and Distributions Examples

Random Variables and Distributions Examples

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Example 1. Consider the experiment of tossing a coin twice.

(a) List the experimental outcomes.

(b) Define a random variable that represents the number of heads occurring on

the two tosses.

(c) Show what value the random variable would assume for each of the

experimental outcomes.

(d) Is this random variable discrete or continuous?

Example 2. Consider the experiment of a worker assembling a product.

(a) Define a random variable that represents the time in minutes required to

assemble the product.

(b) What values may the random variable assume?

(c) Is the random variable discrete or continuous?

Example 3. The probability distribution for the random variable x follows.

x 20 25 30 35

f(x) 0.20 0.15 0.25 0.40

(a) Is this probability distribution valid? Explain.

(b) What is the probability that x = 30? [0.25]

(c) What is the probability that x less than or equal to 25? [0.35]

(d) What is the probability that x is greater than 30? [0.40]

Example 4. The following data were collected by counting the number of operating

rooms in use at a hospital over a 20-day period: On three of the days only one

operating room was used, on five of the days two were used, on eight of the days

three were used, and on four days all four of the hospital’s operating rooms were

used.

(a) Use the relative frequency approach to construct a probability distribution

for the number of operating rooms in use on any given day.

(b) Draw a graph of the probability distribution.

(c) Show that your probability distribution satisfies the required conditions for a

valid discrete probability distribution.


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Example 5. The following table provides a probability distribution for the random

variable x.

x 2 4 7 8

f(x) 0.20 0.30 0.40 0.10

Compute

(a) the expected value of x, E(x). [5.2]

(b) the variance of x, Var(x). [4.56]

(c) the standard deviation of x, Std(x). [2.14]

Example 6. A psychologist determined that the number of sessions required to

obtain the trust of a new patients is either 1, 2, or 3. Let x be a random variable

indicating the number of sessions required to gain the patient’s trust. The following

probability function has been proposed.

6)(

xxf = , for x = 1, 2, or 3.

(a) Is this probability function valid? Explain.

(b) What is the probability that it takes exactly 2 sessions to gain the patient’s

trust? [1/3]

(c) What is the probability that it takes at least 2 sessions to gain the patient’s

trust? [5/6]

(d) Compute the expected value of x, the variance of x, and the standard

deviation of x. [2.333; 0.55556; 0.7454]

Example 7. The following table is a partial probability distribution for a company’s

projected profits (x = profit in $1000s) for the first year of operation (the negative

value denotes a loss).

x −100 0 50 100 150 200

f(x) 0.10 0.20 0.30 0.25 0.10

(a) What is the proper value for f(200)? [0.05]

(b) What is the probability that the company will be profitable? [0.70]

(c) What is the probability that the company will make at least $100 000? [0.40]

(d) What is the expected profit? [55]

(e) What is the variance of the profit? [5 475]


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Example 8. The random variable x is known to be uniformly distributed between

1.0 and 1.5.

(a) Show the graph of the probability density function.

(b) Compute P(x = 1.25). [0]

(c) Compute P(1.0 ≤ x ≤ 1.25). [0.50]

(d) Compute P(1.20 < x < 1.5). [0.60]

Example 9. The probability density of the continuous random variable x is given by

72,5

1)( <<= xxf

(a) Draw its graph and verify that the total under the curve (above the x-axis) is

equal to 1.

(b) Find P(3 < x < 5). [2/5]

Example 10. A university provides bus service to students while they are on

campus. A bus arrives at the University Library and the Student Residence stop

every 30 minutes between 6 A.M. and 11 P.M. during weekdays. Students arrive

at the bus stop at random times. The time that a student waits is uniformly

distributed from 0 to 30 minutes.

(a) Draw a graph of this distribution.

(b) Show that the area of this uniform distribution is 1.00.

(c) What is the mean waiting time? [15]

(d) What is the standard deviation of the waiting times? [8.66]

(e) What is the probability a student will wait more than 25 minutes? [1/6]

(f) What is the probability a student will wait between 10 and 20 minutes? [1/3]

Example 11. Certain coded measurements of the pitch diameter of threads of a

fitting have the probability density

10,)1(

4)(

2<<

+= x

xxf

π.

Find the mean and the variance of this random variable. [0.4413; 0.0785]


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Example 12. For each of the following, determine whether the given values can

serve as the values of a probability distribution of a random variable with the

range x = 1, 2, 3, and 4:

(a) f(1) = 0.25, f(2) = 0.75; f(3) = 0.25, and f(4) = -0.25;

(b) f(1) = 0.15, f(2) = 0.27; f(3) = 0.29, and f(4) = 0.29.

Example 13. For each of the following, determine c so that the function can serve

as the probability distribution of a random variable with the given range:

(a) f(x) = cx, for x = 1, 2, 3, 4, 5; [1/15]

(b) f(x) = c(1/4)x, for x = 1, 2, 3, … [3]

Example 14. For each of the following, determine whether the given values can

serve as the values of a distribution function of a random variable with the range

x = 1, 2, 3, and 4:

(a) F(1) = 0.3, F(2) = 0.5, F(3) = 0.8, and F(4) = 1.2;

(b) F(1) = 0.5, F(2) = 0.4, F(3) = 0.7, and F(4) = 1.0.

Example 15. Find the distribution function of the random variable which has the

probability distribution

5 4, 3, 2, ,1 for ,15

)( == xx

xf

Example 16. If X has the distribution function

≥<≤<≤<≤

<

=

10,1

106,6/5

64,2/1

41,3/1

1,0

)(

x

x

x

x

x

xF

Find

(a) P(2 < X ≤ 6); [1/2]

(b) P(X = 4); [1/6]

(c) the probability distribution of X.


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Example 17. The probability density function (p.d.f) of the random variable X is

given by

<<=elsewhere,0

40,)(

xx

cxf

Find

(a) the value of c; [1/4]

(b) P(X < 1/4) and P(X > 1); [1/4; 1/2]

(c) the distribution function of the random variable X and use it to determine

the probabilities in part (b).

Example 18. The density function of the random variable X is given by

<<−

=elsewhere,0

10),1(6)(

xxxxf

Find

(a) P(X < 1/4) and P(X > 1/2); [5/32; 1/2]

(b) the distribution function of the random variable X and use it to determine

the probabilities in part (a).

Example 19. The distribution function of the random variable X is given by

≥<≤−+

−<=

1,1

11),1(

1,0

)( 21

x

xx

x

xF

Find

(a) P(–1/2 < X < 1/2) and P(2 < X < 3); [1/2; 0]

(b) the probability density of the random variable X and use it to determine the

probabilities in part (a).

Example 20. The distribution function of the random variable X is given by

≥<≤−+

−<=

2,1

22),2(

2,0

)( 41

x

xx

x

xF

Find P(X = –2), P(X = 2), P(–2 < X < 1), and P(0 ≤ X ≤ 2). [0; 0; 3/4; 1/2]

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RV and Distributions Examples