(Solved): Formula Sheet  1. \( Y \) has uniform \( (-1,1) \) distribution. (a) (10 points) What is \( ...

Formula Sheet 

1. \( Y \) has uniform \( (-1,1) \) distribution. (a) (10 points) What is \( \mathrm{E}\left(\left(Y-\mu_{Y}\right)^{4}\right) \) ? (b) (10 points) What is the pdf for \( T=\frac{Y-1}{Y} \) ? Note: \( Y=\frac{1}{1-T} \). 2. Randy has 2 black mice and 7 white mice in his hat. He selects one at random. If the mouse is black, he rolls two fair dice and gets the total. If the mouse is white, he rolls three fair dice and gets the total. (a) (5 points) What is the probability that the total is 4 ? (Express as a fraction - no need to calculate). (b) (5 points) Randy tells Sally that the total is 4 but not which color of mouse he selected. Help her determine the probability that the mouse was black given the information she has. (Again, as a fraction.) 3. (10 points) \( M(t)=e^{t^{2}+e^{t}-1} \) is the moment generating function for a random variable \( W \). What are \( \mathrm{E}(W) \) and \( \operatorname{Var}(W) \) ? 4. (10 points) \( X \sim \) exponential(1) with pdf \( f_{X}(t)=e^{-t}, t \geq 0 \), and \( Y \sim \operatorname{gamma}(2,1) \) with pdf \( f_{Y}(t)=t e^{-t}, t \geq 0 \). Show that their cumulative distribution functions satisfy \( F_{X}(t) \geq F_{Y}(t) \) for all real \( t \). \( \operatorname{binomial}(n, p) \quad 00 \) negative binomial \( (k, p) \quad 00 \quad\left(\begin{array}{c}k+x-1 \\ k-1\end{array}\right) p^{k}(1-p)^{x} \quad x=0,1,2, \ldots \)