L Let $Y$ Be a Bernoulli Random Variable with Success Probability $\operatorname { Pr } ( Y = 1 ) = p$

Essay

L Let $Y$ be a Bernoulli random variable with success probability $\operatorname { Pr } ( Y = 1 ) = p$ , and let $Y _ { 1 } , \ldots , Y _ { n }$ be i.i.d. draws from this distribution. Let $\hat { p }$ be the fraction of successes (1s) in this sample. In large samples, the distribution of $\hat { p }$ will be approximately normal, i.e., $\hat { p }$ is approximately distributed $N \left( p , \frac { p ( 1 - p ) } { n } \right)$ . Now let $X$ be the number of successes and $n$ the sample size. In a sample of 10 voters $( n = 10 )$ , if there are six who vote for candidate $\mathrm { A }$ , then $X = 6$ . Relate $X$ , the number of success, to $\hat { p }$ , the success proportion, or fraction of successes. Next, using your knowledge of linear transformations, derive the distribution of $X$ . 24