STAT 3401, Intro. Prob. Theory 23/27 Jaimie Kwon
1/26/2005
3.4 The binomial probability distribution
♦ Definition 3.6 A “binomial experiment” possesses the following properties:
à A fixed number, n, of trials
à Each trial results in one of two outcomes, Success and Failure (or S and F) [dichotomous or
binary]
à Each trial has the same success probability, p [identical]
à The trials are independent [independence]
♦ Let Y=# of successes observed during n trials
♦ Example 3.5 and 3.6.
Y1 = # of radar units (out of 4) that do not detect the plane
Y2 = # of persons favoring a candidate in a random sample of n=10 voters
Do they follow binomial distribution?
♦ Conditional/unconditional probabilities ;
à The notion of hypergeometric distribution
♦ Definition 3.7 A random variable Y is said to have a “binomial distribution” based on n trials with
success probability p if and only if
p(y) = nCy pyqn-y, y=0,1,…,n and 0≤p≤1
à Histograms? (Use R)
♦ Example 3.7 5000 fuses contain 5% defectives; for a sample of 5 fuses, what’s P(Y≥1)=?
How about P(Y≥4)=?
♦ Notation: Y~bin(n,p)
♦ Theorem 3.7 Let Y~bin(n,p). Then,
µ=E(Y)=np and σ2=V(Y)=npq
à Proof: definitions and ‘binomial’ trick
♦ In practical application, carefully define ‘successes!
♦ HW. Some of the exercises 3.25~49
♦ Keywords: binomial experiment; Y~bin(n,p)