91Ó°ÊÓ

Let X show the number of tails until the first fiveheadsare obtained.

Here, X is a random variable that follows negative binomial distribution that is

\(X \sim NB\left( {r = 5,p = \frac{1}{{30}}} \right)\).

This negative binomial distribution is based on failures.

Thus, the pdf is,

\(f\left( x \right) = {}^{x + r - 1}{C_{r - 1}}{p^r}{\left( {1 - p} \right)^{x - r}},r = 0,1,2 \ldots \)

Since X follows a negative binomial distribution, by its properties, the expectation is:

\(\begin{array}{c}E\left( X \right) = \frac{{r\left( {1 - p} \right)}}{p}\\ = \frac{{5\left( {1 - \frac{1}{{30}}} \right)}}{{\frac{1}{{30}}}}\\ = 145\end{array}\)

As a result, the predicted number of tails gained prior to obtaining five heads is 145.

Since X follows a negative binomial, by its properties, the variance is:

\(\begin{array}{c}V\left( X \right) = \frac{{r\left( {1 - p} \right)}}{{{p^2}}}\\ = \frac{{5\left( {1 - \frac{1}{{30}}} \right)}}{{\frac{1}{{{{30}^2}}}}}\\ = 4350\end{array}\)

As a result, the variation of the number of tails acquired before five tails are achieved is 4350.