p*.
5. Special Discrete Distributions
5.1 Bernoulli and binomial random variables
Def X is a Bernoulli trial(or Bernoulli random variable) with parameter p if sample space S={s, f}, where s is called a success and f a failure and X(s)=1, X(f)=0. The probability function of X is



Bernoulli and binomial random variables
EX = 0xP(X=0)+1xP(X=1)=p
EX2 = 02xP(X=0)+12xP(X=1)=p
Var(X) = EX2 – (EX)2 = p – p2 = p(1-p)



Bernoulli and binomial random variables
Ex 5.1 If in a throw of a fair die the event of obtaining 4, 6 is called a success, and the event of obtaining 1, 2, 3, or 5 is called a failure, then

is a Bernoulli random variable with parameter p=1/3.


Bernoulli and binomial random variables
Let X1, X2, X3, … be a sequence of Bernoulli random variables. If, for all ji=0 or 1, the sequence of events {X1=j1}, {X2=j2}, {X3=j3}, … are independent, we say that {X1, X2, X3, …} and the corresponding Bernoulli trials are independent.


Bernoulli and binomial random variables
Def If n Bernoulli trials all with probability of success p are performed independently, then X, the number of successes, is called a binomial with parameters n and p.
We write as X~B(n,p) in short.
Thm 5.1 X~B(n,p)



Bernoulli and binomial random variables
Ex 5.1 A restaurant serves 8 entrees of fish, 12 of beef, and 10 of poultry. If customers select from these entrees randomly, what is the probability that 2 of the next four customers order fish entrees
Sol:
X~B(4, 8/30=4/15) and
calculate P(X=2)




Bernoulli and binomial random variables

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