Problem 3E

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 3E

Step1 of 3:

Let us consider a random variable X it follows bernoulli distribution with parameter ‘p’.

And range from 0 and 1 with P(0) = p and P(1) = 1 - p

= q.

Here our goal is:

a). We need to sketch the corresponding distribution function.

b). We need to Show that this distribution function has the properties given in Theorem 4.1.

Step2 of 3:

a).

The distribution of X is given by:

The graph of the corresponding distribution function is:

Step3 of 3:

b).

Let

The distribution function of p is 0 everywhere except 0 and 1 thus:

= 0

Similarly,

= 1

Therefore F is a non-decreasing function of X: