Let X be a Poisson random variable with parameter λ.(a) Show thatby using the result of Th...

Let X be a Poisson random variable with parameter λ.

(a) Show that

by using the result of Theoretical Exercise and the relationship between Poisson and binomial random variables.

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(b) Verify the formula in part (a) directly by making use of the expansion of  e-λ+ eλ.

Exercise

Suppose that n independent tosses of a coin having probability p of coming up heads are made. Show that the probability that an even number of heads results is , where q − 1 − p. Do this by proving and then utilizing the identity

where [n/2] is the largest integer less than or equal to n/2. Compare this exercise with Theoretical Exercise 1 of Chapter 3.

Exercise 1

(a) Prove that if E and F arc mutually exclusive, then

(b) Prove that if Ei,i ≥ 1 are mutually exclusive, then