A candidate for public office has claimed that 60% of voters will vote for her. If 5 registered voters were sampled,

- What is the probability distribution function for this voting?
- What is the probability that exactly 3 would say they favor this candidate?
- Calculate the mean and standard deviation of the probability distribution.

### Solution

Let $X$ denote the number of voters who votes for her out of 5 sampled registered voters. A candidate for public office has claimed that 60% of voters will vote for her. That is $p = 0.60$ probability that a randomly selected candidate vote for her.

- Here $n = 5$ and $p=0.60$. The probability distribution of $X$ is Binomial distribution. That is $X\sim B(5,0.60)$.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{5}{x} (0.6)^x (1-0.6)^{5-x},\\ &\qquad \; x=0,1,\cdots, 5. \end{aligned} $$ `

`

- The probability that $X$ is exactly $3$ is

` $$ \begin{aligned} P(X= 3) & =\binom{5}{3} (0.6)^{3} (1-0.6)^{5-3}\\ & = 0.3456\\ \end{aligned} $$ `

The probability that exactly 3 would say they favor this candidate is $P(X=3) = 0.3456$.

- The mean of the probability distribution is $E(X) = np = 5 \times 0.6 = 3$.

The standard deviation of the probability distribution is $sd(X) = \sqrt{np(1-p)} = \sqrt{5 \times 0.6 \times (1- 0.6)} = 1.0954$.