Statistics - Probability Projects

A statistically-minded professor invites you to play the following game, which would determine your grade

A statistically-minded professor invites you to play the following game, which would determine your grade in his course. Given 10 white balls and 10 black balls, you are to distribute all the balls in two boxes, with no box to be left empty. He will then pick one box at random and draw one ball from that box. If the ball is white, you get a 4.0. But if the ball is black, you get a 0.0.
1.1. Describe a ball-allocation scheme that maximizes your chance of getting a 4.0. (5 points)
1.2. Let $A$ be the event that a white ball is picked from a ball-allocation scheme that maximizes your chance of getting a 4.0. Calculate the probability of $A$. (5 points)
Suppose that the joint probability distribution of $X$ and $Y$ is as follows:
$\begin{aligned}
&P(X=-1, Y=0)=0 \\
&P(X=-1, Y=1)=1 / 4 \\
&P(X=0, Y=0)=1 / 6 \\
&P(X=0, Y=1)=0 \\
&P(X=1, Y=0)=1 / 12 \\
&P(X=1, Y=1)=1 / 2
\end{aligned}$
2.1. Find Cov(X, Y) and Correlation(3 X, 2 Y). (Please show the calculations of all moments as well as of all functions of moments that are needed to obtain the magnitudes of Cov(X, Y) and Correlation(3, 2Y).) (20 points)
2.2. Show whether or not $X$ and $Y$ are statistically independent. ( 5 points)
Suppose that the density function of $X$ is given by
$f(x)= \begin{cases}1+x & \text { for }-1<x \leq 0 \\ 1-x & \text { for } 0<x<1 \\ 0 & \text { otherwise }\end{cases}$
3.1. Find the distribution function of $X .(15$ points)
3.2. Find $E(X), E\left(X^{2}\right)$, and $E\left(X^{3}\right)$. (15 points)
3.3. Let $U=X$ and $W=X^{2}$. Find $\operatorname{Cov}(U, W)$. (5 points)
3.4. Are $U$ and $W$ statistically independent? Show why or why not. (2 points)
3.5. Derive the density function of $W=X^{2}$. (12 points)
Let the joint density function of $X$ and $Y$ be given by
$f(x, y)= \begin{cases}\frac{2}{3}(x+2 y) & \text { for } 0<x<1 \text { and } 0<y<1 \\ 0 & \text { otherwise. }\end{cases}$
Find
4.1. $f(x \mid Y=1 / 2)$ (10 points)
4.2. $P(1 / 3<X<2 / 3 \mid Y=1 / 2)$ (5 points)
A multiple-choice quiz consists of eight questions and three offered answers to each question (of which only one is correct). If a student answers each question by tossing a fair die and checking the first answer if he gets a 1 or 2 , the second answer if he gets a 3 or 4 , and the third answer if he gets a 5 or 6 , how many questions would he answer correctly, on average? (8 points)
To reduce the standard deviation of the binomial distribution by half, what change must be made to the number of trials? (5 points)
If a newly-married couple plan to have four children, is it more likely that they will have two boys and two girls or three of one sex and one of the other (i.e., either one boy or three boys)? Assume that the probability of a child being a boy is $1 / 2$ and that the births are independent events.
Please state the probability distribution or density function of your random variable, and show your calculations. ( 10 points)
Let $z_{\alpha}$ be defined by
$\int_{z_{\alpha}}^{\infty} \phi(z) d z=\alpha$ where
$\phi(z)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} z^{2}\right) \quad \text { for }-\infty<z<\infty .$
Find the value of $z_{\alpha}$ for $\alpha=0.05$. (5 points)
Let $X \sim N\left(\mu, \sigma^{2}\right.$, where both $\mu$ and $\sigma^{2}$ are not known. Find the probability of getting a value $x$ that is within one standard deviation of $\mu$. (Hint: Define a linear transformation $Y=X-\mu$. You are asked to find $P(-\sigma<Y<\sigma)$.) (5 points)