o7/10/2010
Discrete Probability Distributions
Binomial Distribution Poisson Distribution Hypergeometric Distribution
• Consider a random experiment having only two outcomes : • success (S) with probability p, and failure (F) with probability 1 − p. • Define a random variable X which takes the value 1 if S occurs and 0 when F occurs, i.e.,
with probability p ith b bilit ⎧1, X =⎨ ⎩0, with probability 1 − p
probability mass function of X is for x = 1 ⎧ p, p( x) = ⎨ ⎩1 − p, for x = 0
• In functional form, the pmf of X is
p ( x) = p x (1 − p )1− x , x = 0,1.
• Then X is said to have Bernoulli distribution.
QAM - I by Prof. Gaurav Garg (IIM Lucknow) QAM - I by Prof. Gaurav Garg (IIM Lucknow)
• A discrete random variable X is said to have a Bernoulli Distribution with parameter p (0≤ p ≤ 1 ), if its probability mass function is given by
• Where q = 1 – p, and we write
• E(X)= p (show) • Var(X)= p(1‐p) (show)
QAM - I by Prof. Gaurav Garg (IIM Lucknow)
• Consider the random experiment of tossing a fair die. • Let the event of getting “6” is considered as a success. • Getting any other number is considered as a failure. • Probability of success in a toss is 1/6. • Probability of failure in a toss is 5/6. • If we repeat the tosses 3 times, i.e., • If we have 3 trials. • The probability of success and failure will not change. • Trials are independent.
QAM - I by Prof. Gaurav Garg (IIM Lucknow)
• Let X is a random variable which counts the number of successes. • X could take the values 0, 1, 2, or 3. • P(X=0) = P(No Success in 3 trials) = (5/6)(5/6)(5/6) = (5/6)3 • P(X=1) = P(1 success in 3 trials) = P (SFF or FSF or FFS) = P(SFF) + P(FSF) + P(FFS) = P(S)P(F)P(F) + P(F)P(S)P(F) ( ) ( ) ( ) ( ) ( ) ( ) + P(F)P(F)P(S) ( ) ( ) ( ) =(1/6)(5/6)(5/6) + (5/6)(1/6)(5/6) + (5/6)(5/6)(1/6) = 3. (1/6)1 (5/6)2 • P(X=2) = P(2 Successes in 3 trials) = P( SSF or SFS or FSS) = 3. (1/6)2 (5/6)1 •...