Question

Question Two :
A discrete random variable X has negative binomial distribution,
with 6 successes required each with probability of success 0.4
Determine the value of ...
a) ...E(X).
b) ... Var( X)
c) ... P ( X = 12).

Answer 1

Using the negative binomial distribution, it is found that:

a) $E(X) = 15$

b) $Var(X) = 6.67$

c) $P(X = 12) = 0.0883$

Negative binomial distribution:

It is the number of trials until q successes of a binomial variable, with p probability of success.

The expected value is:

$E(X) = \frac{q}{p}$

The variance is:

$Var(X) = \frac{pq}{(1 - p)^2}$

The probability mass function is:

$P(X = x) = C_{x+q-1,q-1}(1 - p)^xp^q$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

In this problem, the parameters are: $q = 6, p = 0.4$.

Item a:

$E(X) = \frac{6}{0.4} = 15$

Item b:

$Var(X) = \frac{6(0.4)}{(1 - 0.4)^2} = 6.67$

Item c:

$P(X = 12) = C_{17,5}(0.6)^12(0.4)^6 = 0.0883$

To learn more about the negative binomial distribution, you can take a look at https://brainly.com/question/15246027