Chapter 6, Problem 6.26STPE

Let $X_1,\mathrm{...},X_n$, be independent nonnegative integer valued random variables, and let $a_i=P\left(X_i\text{ is even}\right),i=1,\mathrm{...},n$. With $S={\displaystyle {\sum }_{=1}^n{X}_i}$ we want to determine $p=P\left(S\text{ is even}\right)$. Let $Y_i=1$ if $X_i$ is even and let it equal -1 is odd. In parts (a) and (b) fill in the missing word at the end of the sentence.

a. S is even if and only if the number of $X_1,\mathrm{...},X_n$ that are odd is

b. S is even if and only if $\left[{\displaystyle {\prod }_{i=1}^n{Y}_i}\right]$ is

c. Find $E\left[{\displaystyle {\prod }_{i=1}^n{Y}_i}\right]$.

d. Find $P\left(S\text{ is even}\right)$.

Hint: Use parts (b) and (c).