Page 83 - Probability and Statistical Inference

60 1. Notions of Probability

ution, 0 < p < 1. Show that

{Hint: Use (1.4.3), (1.7.2) and direct calculations.}
1.7.4 Suppose that a random variable X has the Binomial(n, p) distribution,
0 < p < 1. For all x = 0, 1, ..., n, show that

This recursive relation helps enormously in computing the binomial probabili-
ties successively for all n, particularly when n is large. {Hint: Use (1.7.2) and
direct calculations.}
1.7.5 (Exercise 1.7.4 Continued) Suppose that a random variable X has the
Binomial(n, p) distribution, 0 < p < 1. Find the value of x at which P(X = x) is
maximized. {Hint: Use the result from the Exercise 1.7.4 to order the prob-
abilities P(X = x) and P(X = x + 1) for each x first.}
1.7.6 The switchboard rings at a service desk according to a Poisson distri-
bution on the average five (= λ) times in a ten minute interval. What is the
probability that during a ten minute interval, the service desk will receive
(i) no more than three calls?
(ii) at least two calls?
(iii) exactly five calls?
1.7.7 Suppose that a random variable X has the Poisson (λ) distribution, 0
< λ < ∞. Show that

{Hint: Use (1.4.3), (1.7.4) and direct calculations.}
1.7.8 Suppose that a random variable X has the Poisson (λ) distribution, 0
< λ < ∞. For all x = 0, 1, 2, ..., show that

This recursive relation helps enormously in computing the Poisson probabili-
ties successively for all x, particularly when x is large. {Hint: Use (1.7.4) and
direct calculations.}