Let X ~ Bin(n, p).

a. Show that if x is an integer between 1 and n inclusive,then

b. Show that if X ~ Bin(n, p), the most probable value for X is the greatest integer less than or equal to np + p. [Hint: Use part (a) to show that P(X = x) ≥ P(X = x - 1) if and only if x ≤ np + p.]

Answer:

Step 1 of 2:

(a)

In this question, we a asked to prove the following.

…………(1)

Where x is an integer between 1 and n inclusive and is probability.

If is a random variable whose distribution is binomial with parameter and , then we can write

The probability mass function of a binomial random variable is defined by,

= otherwise

………………..(2)

Replace with in equation (2), we get

………………….(3)

=

(can be written as

can be written as

simplify the equation

=

= (

= (

Hence proved.