3 2014-Spring
3.2 Problem 1
Define
\[I_i = \begin{cases} 1 & \text{Game $i$ and Game $i+1$ are won},\\ 0 & \text{otherwise} \end{cases}\]
Thus, \[ \begin{align*} R &= \sum_{i=1}^{m-1}I_i\\ ER &= \sum_{i=1}^{m-1}EI_i\\ Var(R) &= \sum_{i=1}^{m-1}Var(I_i) + 2\sum_{i < j}Cov(I_i,U) \end{align*} \]
3.2.1 Part 1.a
\[ \begin{align*} EI_i &= P(I_i=1) \\ &= a_ia_{i+1}\\ ER &= \sum_i EI_i\\ &=\sum_{i=1}^{m-1}a_ia_{i+1} \end{align*} \]
\[ \begin{align*} Var(I_i) &= E[I_i^2]-E[I_i]E[I_i]\\ &= E[I_i](1-E[I_i])\\ &= a_ia_{i+1}(1-a_ia_{i+1}) \end{align*} \]
\[ \begin{align*} Cov(I_i,I_j) &= \begin{cases} 0 & j-i\geq 2(i<j)\\ E[I_{i}I_{i+1}]-E[I_i]E[I_{i+1}] & \text{otherwise}\\ \end{cases}\\ &= \begin{cases} 0 & j-i\geq 2(i<j)\\ a_ia_{i+1}a_{i+2}(1-a_{i+1}) & \text{otherwise}\\ \end{cases}\\ \end{align*} \]
Thus, \[ Var(R) = \sum_{i=1}^{m-1}a_ia_{i+1} + 2\sum_{i=1}^{m-2} a_ia_{i+1}a_{i+2}(1-a_{i+1}) \]
3.2.2 Part 1.b
When \(a_n=p=0.1\)
\[ \begin{align*} ER &= \sum_{i=1}^{m-1}p^2\\ &= (m-1)p^2\\ Var(R) &= \sum_{i=1}^{m-1}a_ia_{i+1} + 2\sum_{i=1}^{m-2} a_ia_{i+1}a_{i+2}(1-a_{i+1})\\ &= (m-1)p^2+2(m-2)p^3(1-p) \end{align*} \]
3.3 Problem 2
\[f(x) = x/2 \text{ for } 0 < x <2\]
3.3.1 Part 2.a
\(S=X_1+X_2\) Using convolution theorem:
\[ \begin{align*} f_S(s) &= \int \frac{x}{2} \frac{s-x}{2} dx\\ &0 < s-x < 2\\ & 0 <x<2\\ &\implies s-2 < x < s \text{ and } 0 < x < 2\\ &\text{Case 1: } 0 < s < 2\ f_S(s) = \int_0^s \frac{x}{2} \frac{s-x}{2} dx = \frac{s^3}{24}\\ &\text{ Case 2: } 2 < s < 4\ f_S(s) = \int_{s-2}^4 \frac{x}{2} \frac{s-x}{2} dx = \frac{1}{4}(\frac{s(4s-s^2)}{2}-\frac{(4^3-(s-2)^3)}{3})\\ \end{align*} \]
3.3.2 Part 2.b
\[L = min(X_1, X_2, X_3, \dots X_{100})\]
$F_L = 1 - P(min(X_i) y) = 1 - _i P(X_i y) = 1-(1-P(X_iy)) = 1- (1-)^n $
3.3.3 Part 2.c
\[R=X_1/X_2\]
\[ \begin{align*} P(X_1/X_2 \leq z) &= P(X_1 \leq zX_2)\\ &= \int_0^{min(zx_2,2)} \frac{z^2x_2^2}{4}x_2 dx_2 \end{align*} \]