STATISTICS 301 TA: Perla E. Reyes DISCUSSION 5 Pag. 1
Review
1. 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:
(i) Each trial results in one of two possible outcomes, which for convenience are labeled
success and failure.
(ii) The probability of obtaining a success remains constant from trial to trial. This constant
probability of success is denoted by the number πor p. The probability of a failure is
denoted by (1 −π) or q.
(iii) The trials are independent.
2. 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) = π, P (Xi= 0) = 1 −π, 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)!πx(1 −π)n−xfor x= 0,1,·· · , n.
Then we say that the random variable Xhas the binomial distribution, denoted by
X∼Bin(n, π).
(i) The mean of binomial distribution is
µ=nπ
(ii) The standard deviation of binomial distribution is
σ=pnπ(1 −π)
Practice Problem
1. According to the Mendelian theory of inherited characteristics, a cross fertilization of related
species of red and white-flowered plants produces an offspring generation which contain 25%
red-flowered plants. Suppose that a horticulturist wishes to cross 5 pairs of the cross-fertilized
species. Of the 5 offspring, what is the probability that
(a) There will be one red-flowered plants?
(b) There will be 1 or more red-flowered plants?