DISCUSSION 5
(Feb. 21. 2004)
Properties for calculating probability
•P(A∩Bc) = P(A)−P(A∩B).
P(Ac∩B) = P(B)−P(A∩B).
•P(A∩B) = P(A)P(B|A) = P(B)P(A|B)
•P(Bc|A) = 1 −P(B|A)
Bayes’ Formula
For i=1,2,...,n, the posterir probabilities are P(Bi|A) = P(Bi∩A)
P(A)=P(A|Bi)∗P(Bi)
PP(A|Bi)∗P(Bi)
Bernoulli Trials
Suppose we observe a sequence of ntrials. Let Xndenote the outcome of the nth trial. The trials
are called Bernoulli Trial if the following assumptions are satisfied:
1. Each trial results in one of two possible outcomes, which for convenience are labeled success and
failure.
2. The probability of obtaining a success remains constant from trial to trial. This constant probability
of success is denoted by the number p. The probability of a failure is denoted by q.
3. The trials are independent.
The Binomial Distribution
If random variables X1, X2,···, Xnare Bernoulli trials,
Xi=1 success in ith trial
0 failure in ith trial
P(Xi= 1) = p, P (Xi= 0) = q(= 1 −p), i = 1,2,···, n.
Let X=X1+X2+···+Xn, which is the total number of successes. The sampling distribution
of Xis given by
P(X=x) = n!
x!(n−x)!pxqn−xfor x= 0,1,···, n.
Then we say that the random variable Xhas the binomial distribution, denoted by
X∼Bin(n, p).
1. The mean of binomial distribution is
µ=np
2. The standard deviation of binomial distribution is
σ=√npq, where q= 1 −p.
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