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A38

A P P E N D I X C

Probability and Probability Distributions

Probability Distributions

If each outcome in a sample space is equally likely, then the probability of is

S

x

given by

Frequency of x

n x

P x

Number of outcomes in S

n S

where

n S

is the number of outcomes in the sample space. Note that if the range

of a discrete random variable consists of

m

different values

x

, x

, x

, . . . , x

,

1

2

3

m

then the sum of the frequencies of the

x

is equal to

n S .

This can be written as

i

. . .

n x

n x

n x

n x

n S .

1

2

3

m

The collection of probabilities corresponding to the values of the random variable

is the probability distribution of the random variable. For example, the proba-

bility distribution for the random variable representing the number of heads that

turn up when three coins are tossed and a corresponding graphical representation

are shown in Figure C.21.

Probability Distribution

P(x)

3

8

1

4

1

8

x

Random variable, x

0

1

2

3

0

1

2

3

1

3

3

1

Random variable

Probability of x, P x

8

8

8

8

F I G U R E C . 2 1

An important fact about the probability distribution shown is that the sum of

the probabilities of the values of the random variable is 1. So,

1

3

3

1

P 0

P 1

P 2

P 3

1.

8

8

8

8

This is true of the probability distribution for any discrete random variable.

Probability Distribution for a Discrete Random Variable

If the range of a discrete random variable consists of

m

different values

x

, x

, x

, . . . , x

1

2

3

m

then the sum of the probabilities of

x

is equal to 1. This can be written as

i

. . .

P x

P x

P x

P x

1.

1

2

3

m

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