Fail- ure. Success occurs with probability p and failure with proba ...

University of California, Los Angeles

Department of Statistics

Statistics 403

Instructor: Nicolas Christou

Some special discrete probability distributions

? Bernoulli random variable:

It is a variable that has 2 possible outcomes: ¡°success¡±, or ¡°failure¡±. Success occurs with probability p and failure with probability 1 ? p.

1

? Binomial probability distribution:

Suppose that n independent Bernoulli trials each one having

probability of success p are to be performed. Let X be the

number of successes among the n trials. We say that X follows

the binomial probability distribution with parameters n, p.

Probability mass function of X:

?

?

n? x

n?x

? ?p (1 ? p)

P (X = x) = ?

,

x

x = 0, 1, 2, 3, ¡¤ ¡¤ ¡¤ , n

or

P (X = x) = nCx px(1 ? p)n?x,

x = 0, 1, 2, 3, ¡¤ ¡¤ ¡¤ , n

where

?

?

n!

n?

? ? =

nCx = ?

x

(n ? x)!x!

Expected value of X:

E(X) = np

Variance of X:

¦Ò 2 = np(1 ? p)

r

Standard deviation of X: ¦Ò = np(1 ? p

2

? Geometric probability distribution:

Suppose that repeated independent Bernoulli trials each one having probability of success p are to be performed. Let X be the

number of trials needed until the first success occurs. We say

that X follows the geometric probability distribution with parameter p.

Probability mass function of X:

P (X = x) = (1 ? p)x?1p,

x = 1, 2, 3, ¡¤ ¡¤ ¡¤

Expected value of X:

E(X) =

Variance of X:

¦Ò2 =

1?p

p2

s

Standard deviation of X: ¦Ò =

3

1

p

1?p

p2

? More on geometric probability distribution ¡¤ ¡¤ ¡¤

Repeated Bernoulli trials are performed until the first success

occurs. Find the probability that

¨C the first success occurs after the kth trial

¨C the first success occurs on or after the kth trial

¨C the first success occurs before the kth trial

¨C the first success occurs on or before the kth trial

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? Negative binomial probability distribution:

Suppose that repeated Bernoulli trials are performed until r successes occur. The number of trials required X, follows the so

called negative binomial probability distribution.

Probability mass function of X is:

?

?

?

?

x ? 1? r?1

?

?p

P (X = x) = ?

(1 ? p)x?r p, or

r?1

x ? 1? r

x?r

?

?p (1 ? p)

P (X = x) = ?

r?1

x = r, r + 1, r + 2, ¡¤ ¡¤ ¡¤

r

p

r 1?p

p2

Expected value of X:

E(X) =

Variance of X:

¦Ò2 =

s

Standard deviation of X: ¦Ò = r 1?p

p2

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