3. (30 pts) Consider a biased coin, whose probability of turning up heads is $p$. Let $X$ be the number of flips until the first head. Then from Lesson 5, $X \sim \text{Geometric}(p)$ and $p_X(k) = P(X = k) = (1 - p)^{k-1}p$, $k = 1, 2, 3, \dots$ For this problem, the following formulas are helpful ($|q| < 1$): \begin{align*} \sum_{k=0}^{\infty} q^k &= \frac{1}{1 - q}, \quad \sum_{k=0}^n q^k = \frac{1 - q^{n+1}}{1 - q}, \quad \sum_{k=1}^{\infty} kq^{k-1} = \frac{1}{(1 - q)^2} \end{align*} a) (6 pts) Show that $\sum_{k=1}^{\infty} p_X(k) = 1$. b) (6 pts) Find $E(X)$. c) (4 pts) Let $Y$ be the number of flips until the second tail. Find the pdf of $Y$. d) (4 pts) Find $E(Y)$. Hint: Let $Y$ be a sum of geometric variables. e) (4 pts) Let $W$ be the number of flips before the first head (not counting the first head). Find the pdf of $W$. f) (6 pts) Let $V \sim \text{Geometric}(0.5)$, find the pdf and cdf of $V$.
