Q$3.$ Just part a$)$

Q$4.$ Part a and b

Suppose that, for $-11,$ the random variables $x$ and $Y$ have the joint probability

density function given by

$f(x,y)=\{\begin{array}{ccc}[1-(1-2e^{-x})(1-2e^{-y})]e^{-x-y}& ifx0\text{a}\text{n}\text{d}& y0,\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}\text{.}& \end{array}$

$($a$)$ Prove that the function $f(x,y)$ is indeed a joint probability density function for

all $in[-1,1].$

$($b$)$ Show that the marginal distributions of $x$ and $Y$ are both exponential distribu$-$

tions with mean $=1.$

$($c$)$ Show that $x$ and $Y$ are independent if and only if $=0.$

$($d$)$ Find the covariance Cov$(x,Y)$ of $x$ and $Y.$

$($e$)$ Show that Cov$(x,Y)=0$ if and only if $=0.$

$(15$ points$)$

Suppose that $x$ is a binomial random variable based on $n$ Bernoulli trials with success

probability $p$ and consider the random variable defined by

$Y=n-x$

$($a$)$ Prove that for $y=0,1,$dots,$n,$ we have

$P(Y=y)=([n],[n-y])p^{n-y}(1-p{)}^y=([n],[y])(1-p{)}^y{p}^{n-y}.$

$($b$)$ Use the result in $($a$)$ to find the probability distribution of the random variable

$Y$ and specify its parameters.

$($c$)$ The probability that a patient recovers from a rare blood disease is $0\mathrm{.}75.$ If $15$

people are known to have contracted this disease, what is the probability that

$()$ at least $10$ survive,

$()$ from $3$ to $8$ survive, and

$()$ exactly $5$ survive?

$[$Hint: Use Appendix Table A$.1$ in the textbook!$]$