Answers to Practice problems on Chapter 3.

1(a) G takes values 2,4,6,8,. P(G=2n)= 1/2^n. (b) E(G) = 4.

2(a) Let X take values 1 with probability 1/2, -1 with probability 1/2. and Y=-X take values -1 1/2 , 1 with probability 1/2. Then$XY = -X^2$ takes values -1 with probability 1. E(X) = 0, E(-X) = 0, $E(-X^2) = -1.$
(b) Assume X and Y are discrete and take values $x_1, \ldots, x_n$ and $y_1, \ldots y_n.$ Therefore $XY$ takes values $x_iy_k, i,k=1,\ldots,n.$

\begin{eqnarray*}E(XY) &= \sum_{i=1,k=1}^{n} x_iy_k P(XY= x_iy_k)\\ &= \sum_{i... ...sum_{i=1}^n x_i P(X=x_i) \sum_{k=1}^n y_k P(Y=y_k)\\ &= E(X)E(Y) \end{eqnarray*}



3. Let X take values 1 with probability 1/2, -1 with probability 1/2. $g(x)=X^2$.

4. Let $X_k$ be the number of mistakes on page k. Then $X_k$ is possion(1). (a) Find $P(X_1 \geq 2)$ (b) Since $X_1, \ldots X_5$ are all indepenent $X_1 + \ldots + X_5$ is poisson (5) (why ?). Find P( $X_1 + \ldots + X_5$ >2) http://www.math.ubc.ca/~athreya/302/sol
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