Solutions - 5100-02

(a)

Let $X_i$ be A’s winnings after the $i^{\text{th}}$ round. Since A either loses or gains one dollar each round, $X_i\in \{\pm 1\}$. We have:

$$X=\sum _{i=1}^5{X}_i$$

Therefore $X\in \{-5,-3,-1,1,3,5\}$.

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(b)

(1) Describe the PMF of $X$:

$X=5$ if and only if all 5 rounds are tails, so $P[X=5]=\frac{1}{2^5}=\frac{1}{32}$.

$X=-5$ if and only if all 5 rounds are heads, so $P[X=-5]=\frac{1}{32}$.

$X=3$ if 4 rounds are tails, and $X=-3$ if 4 rounds are heads. Since the coin is fair, $P[X=3]=P[X=-3]=\left(\genfrac{}{}{0px}{}{5}{4}\right){(0.5)}^5=\frac{5}{32}$.

$X=1$ if 3 rounds are tails, and $X=-1$ if 3 rounds are heads. Since the coin is fair, $P[X=1]=P[X=-1]=\left(\genfrac{}{}{0px}{}{5}{3}\right){(0.5)}^5=\frac{10}{32}$.

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(2) Define the PMF of $X$:

$$P_X(x)=\{\begin{array}{cc}\frac{1}{32}& x=-5\\ \frac{5}{32}& x=-3\\ \frac{10}{32}& x=-1\\ \frac{10}{32}& x=1\\ \frac{5}{32}& x=3\\ \frac{1}{32}& x=5\\ 0& \text{otherwise}\end{array}$$

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(3) Define the CDF of $X$:

Note that the jumps in the CDF occur at the possible discrete values of $X$.

$$F_X(x)=\{\begin{array}{cc}0& x<-5\\ \frac{1}{32}& -5\le x<-3\\ \frac{6}{32}& -3\le x<-1\\ \frac{16}{32}& -1\le x<1\\ \frac{26}{32}& 1\le x<3\\ \frac{31}{32}& 3\le x<5\\ 1& 5\le x\end{array}$$