Problem 4E

Let Y be a binomial random variable with n = 1 and success probability p.

a Find the probability and distribution function for Y .

b Compare the distribution function from part (a) with that in Exercise 4.3(a). What do you conclude?

Reference

A Bernoulli random variable is one that assumes only two values, 0 and 1 with p(1) = p and p(0) = 1 − p ≡ q.

a Sketch the corresponding distribution function.

b Show that this distribution function has the properties given in Theorem 4.1.

Reference

Solution:

Step 1 of 2:

Let Y be a Binomial random variable with n = 1, p be the probability of success and q be the probability of failure

The claim is to find the probability and distribution function for Y.

The probability for Y is given by

P(Y = K) =

we have n = 1

Then, P(Y = K) =

If k =o, P(Y = K) = 1 - p

If k =1, P(Y = K) = p

The probability function of Y is

Hence, the distribution function is