Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $\$2$ for each black ball selected and we lose $\$1$ for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed $n$ times. What are the possible values of X ?
Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares her number with that of player 3, and so on. Let X denote the number of times player 1 is a winner. Find $P(X=i)$ for $i =0,1,2,3,4$.
Suppose that the distribution function $X$ of a random variable is given by $$ F_X(x)= \begin{cases} 0,& x<0,\\ \frac{x}{4},& 0\leq x<1,\\ \frac{1}{2}+\frac{x-1}{4},& 1\leq x<2,\\ \frac{11}{12},&2\leq x<3,\\ 1,&3\leq x. \end{cases} $$ Find $P(X=i)$ for $i=1,2,3$ and $P\bigl(\frac{1}{2}<X<\frac{3}{2}\bigl)$.
Suppose that two teams play a series of games that ends when one of them has won $i$ games. Suppose that each game played is, independently, won by team A with probability $p$. Find the expected number of games that are played when $i=2$ and when $i=3$. Also, show that this number is maximized when $p=0.5$ in both cases.
You have $\$ 1000$, and a certain commodity presently sells for $\$ 2$
per ounce. Suppose that after one week the commodity will sell for either
$\$1$ or $\$4$ an ounce, with these two possibilities being equally
likely.
a. If your objective is to maximize the expected amount of money that
you possess at the end of the week, what strategy should you employ?
b. If your objective is to maximize the expected amount of the
commodity that you possess at the end of the week, what strategy
should you employ?
A box contains 5 red and 5 blue marbles. Two marbles are withdrawn
randomly. If they are of the same color, then you win $\$1.10$. If they are
of different colors, then you loose $\$1.00$. Calculate
a. the expected value of the amount you win.
b. the variance of the amount you win.
If $X$ has distribution function $F_X,$ what is the distribution function $F_Y$ of $Y=e^X$?
Let X be a random variable having expected value $\mu$ and variance $ \sigma^2$. Find the expected value and variance of $$ Y=\frac{X-\mu}{\sigma}. $$
Given $a,b\in \mathbb{R}$, find $ \operatorname{Var}(X )$ if $$ P(X=a)=p=1-P(X=b). $$