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
4
? 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
5
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- lecture 4 random variables and distributions university of washington
- reading 6a expectation variance and standard deviation for continuous
- normal dist 1 normal dist 2 colorado state university
- normal distribution university of california los angeles
- normal distribution university of notre dame
- distribution functions ti83 portland state university
- chapter 5 review the normal probability and standardization provide an
- probability distributions duke university
- variance and standard deviation university of pennsylvania
- chapter 3 probability distributions the hong kong polytechnic
Related searches
- 4 p of marketing with example
- s p 500 stocks with dividends
- p value calculator with significance level
- tail probability p table
- heart failure with preserved ejection fraction
- heart failure with reduced ejection fraction
- coding respiratory failure with copd
- p value calculator with graph
- icd 10 heart failure with preserved ejection
- standard deviation with probability excel
- heart failure with preserved ejection icd 10
- chemotherapy success and failure rates