CHAPTER 4 4 4.1 - Discrete Models G eneral distributions C lassical: Binomial, Poisson, etc. 4 4.2 - Continuous Models G eneral distributions C

CHAPTER 4

• 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc.

• 4.2 - Continuous Models General distributions Classical: Normal, etc.

X

1

6

1

6

1

6

1

6

1

6

1

6

Motivation ~ Consider the following discrete random variable…

2

Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”

Probability Table

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1

Probability Histogram

“What is the probability of rolling a 4?”

( 4)P X

X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6.

Total Area = 1

P(X = x)

Den

sit

y

(4)f

X

1

6

1

6

1

6

1

6

1

6

1

6

3


Probability Table

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1



( 4)P X 1

6


Total Area = 1

P(X = x)

Motivation ~ Consider the following discrete random variable…D

ensi

ty

(4)f


4


P(X = x)

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1


Cumulative distribution

P(X x)

F(x)

1/6

2/6

3/6

4/6

5/6

1


5


P(X = x)

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1


Cumulative distribution

P(X x)

F(x)

1/6

2/6

3/6

4/6

5/6

1

“staircase graph” from 0 to 1

Time intervals = 0.5 secsTime intervals = 2.0 secsTime intervals = 1.0 secsTime intervals = 1.0 secsTime intervals = 5.0 secs

“In the limit…”

POPULATION

random variable XContinuous

6

Example: X = “reaction time”

“Pain Threshold” Experiment:Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn.

In principle, as # individuals in samples increase without bound, the class interval widths can be made arbitrarily small, i.e, the scale at which X is measured can be made arbitrarily fine, since it is continuous.

SAMPLE

Total Area = 1

we obtain a density curve

x

7


x

Cumulative probability F(x) = P(X x) = Area under density curve up to x

f(x) no longer represents the probability P(X = x), as it did for discrete variables X.

• f(x) 0• Area = 1

f(x) = density function

00

x

In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)

However,

F(x) increases continuously from 0 to 1.

we can define “interval probabilities” of the form P(a X b), using F(x).



8

“In the limit…”Cumulative probability F(x) = P(X x)

= Area under density curve up to x

• f(x) 0• Area = 1



a b a b

However, we can define “interval probabilities” of the form P(a X b), using F(x).

F(a)

F(b)

F(b) F(a)


In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)

An “interval probability” P(a X b) can be calculated as the amount of area under the curve f(x) between a and b, or the difference P(X b) P(X a), i.e., F(b) F(a). (Ordinarily, finding the area under a general curve requires calculus techniques… unless the “curve” is a straight line, for instance. Examples to follow…)


9

“In the limit…”Cumulative probability F(x) = P(X x)

= Area under density curve up to x

• f(x) 0• Area = 1


a b a b


F(a)

F(b)

F(b) F(a)


Moreover, and . ( ) ( ) .

2 2x f x dx( )

x f x dx

10



Thus, in general, P(a X b) = = F(b) F(a). ( )bf x dx

a


• f(x) 0• Area = 1( )

1f x dx


Fundamental Theorem of

Calculus


X

1

6

1

6

1

6

1

6

1

6

1

6

Consider the following continuous random variable…

11



( 4)P X 1 6


Total Area = 1

Probability Table

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1

P(X = x)

F(x)

1/6

2/6

3/6

4/6

5/6

1

Cumul Prob

P(X x)


Den

sit

y

X

1

6

1

6

1

6

1

6

1

6

1

6


12

Example: X = “Ages of children from 1 year old to 6 years old”


( 4)P X

Further suppose that X is uniformly distributed over the interval [1, 6].


Total Area = 1

Probability Table

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1

P(X = x)

F(x)

1/6

2/6

3/6

4/6

5/6

1

Cumul Prob

P(X x)


Den

sit

y

1 6

F(x)

1/6

2/6

3/6

4/6

5/6

1

Probability Table

x f(x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

1

1

6

1

6

1

6

1

6

1

6

1

6

X


13



( 4)P X



Total Area = 1

P(X = x)Cumul Prob

P(X x)


Den

sit

y

1 6

1

6

1

6

1

6

1

6

1

6

1

6

X





Total Area = 1

Cumul Prob

P(X x)

that a random child is 4 years old?”

( ) 0.20f x

Check? Base = 6 – 1 = 5

Height = 0.25 0.2 = 1

doesn’t mean…..

= 0 !!!!!

> 0

The probability that a continuous random variable is exactly equal to any single value is ZERO!

Den

sit

y

A single value is one point out of an infinite continuum of points on the real number line.

( 4)P X 1 6( 4.000000000......)P X

F(x)

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X




( 4)P X


Cumul Prob

P(X x)

that a random child is 4 years old?”

( ) 0.20f x

actually means....

= (5 – 4)(0.2) = 0.2 (4 5)P X between 4 and 5 years old?”

NOTE: Since P(X = 5) = 0, no change for P(4 X 5), P(4 < X 5), or P(4 < X < 5).

Den

sit

y

Alternate way using cumulative distribution

function (cdf)…

( 5)P X

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X





Cumul Prob

P(X x)

that a random child is

( ) 0.20f x

under 5 years old?

Den

sit

y

0.2 (5 1) 0.8(5)F


function (cdf)…

( 4)P X

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X





Cumul Prob

P(X x)


( ) 0.20f x

under 4 years old?

Den

sit

y

0.2 (4 1) 0.6(4)F


function (cdf)…

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X





Cumul Prob

P(X x)


( ) 0.20f x

Den

sit

y

between 4 and 5 years old?”

(4 5)P X ( 5)P X ( 4)P X


function (cdf)…

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X





Cumul Prob

P(X x)


( ) 0.20f x

Den

sit

y

between 4 and 5 years old?”

(4 5)P X = F(5)

( 5)P X ( 4)P X F(4)


function (cdf)…

= 0.8 – 0.6 = 0.2

F(x)

1

6

1

6

1

6

1

6

1

6

1

6

X




Cumul Prob

P(X x)( ) 0.20f x


x

For any x, the area under the curve is

F(x) = 0.2 (x – 1).

Den

sit

y

1

6

1

6

1

6

1

6

1

6

1

6




( ) 0.20f x

x

For any x, the area under the curve is

F(x) = 0.2 (x – 1).

Den

sit

y


F(x) = 0.2 (x – 1)


(compare with “staircase graph” for discrete case)

1

6

1

6

1

6

1

6

1

6

1

6




( ) 0.20f x

Den

sit

y


F(x) = 0.2 (x – 1)

F(4) = 0.6

F(5) = 0.8

“What is the probability that a child is between 4 and 5?”

(4 5)P X = F(5)

( 5)P X ( 4)P X F(4) = 0.8 – 0.6 = 0.2

0.2




( ) .08 ( 1)f x x

Den

sit

y

> 0

Area = 1

2Base Height(6 1) (0.4)

= 1 0.4



( ) .08 ( 1)f x x

“What is the probability that a child is under 4 years old?” ( 4)P X

Den

sit

y

Area = 1

2Base Height(4 1) ???(4)f.08 (4 1)

0.4

( 4)P X



( ) .08 ( 1)f x x

Den

sit

y

Area = 1

2Base (4 1)

= 0.36

0.24

Alternate method, without having to use f(x):

Use proportions via similar triangles.

h = ? 4 1

h

0.4

6 1

0.4

0.24h 0.36

“What is the probability that a child is under 4 years old?” ( 4)P X 0.36



( ) .08 ( 1)f x x

“What is the probability that a child is under 4 years old?”

Den

sit

y

“What is the probability that a child is over 4 years old?” ( 4)P X 1 0.36 0.64

0.36

0.64

( 4)P X 0.36



( ) .08 ( 1)f x x


F(x) = ????????

“What is the probability that a child is under 4 years old?” ( 4)P X

Exercise…

Den

sit

y

x

?“What is the probability that a child is under 5 years old?”

“What is the probability that a child is between 4 and 5?”

( 5) (5)P X F (4 5)P X

(4)F

Unfortunately, the cumulative area (i.e., probability) under most curves either… requires “integral calculus,” or is numerically approximated and tabulated.

28

IMPORTANT SPECIAL CASE: “Bell Curve”

Documents

CHAPTER 4 4 4.1 - Discrete Models G eneral distributions C lassical: Binomial, Poisson, etc. 4 4.2 - Continuous Models G eneral distributions C