| Concept | Examples | Page Number |
|---|---|---|
| Expected Value | 1.18, 1.20, 1.21, 1.22 | 17-19 |
| Expected Value of Sums | 6.8, 6.9, 6.10, 6.11, 6.12 | 197-200 |
| Variance and Covariance | 1.26, 1.27, 1.29, | 23-25 |
| Concept | Questions | Page Number |
|---|---|---|
| Expected Value | 40-49 | 29-30 |
| Expected Value of Sums | 1-9 | 223 |
| Variance and Covariance | (i) 50-56; (ii) 10-16 | (i) 30-31; (ii) 223-224 |
A gambler plays roulette and bets \$1 on black 19 times. They wins \$1 with probability 18/38 and loses \$1 with probability 20/38. What are the expected winnings?
You want to invent a gambling game in which a person pays \$1 to enter and rolls two dice. If the sum is $7$, the person earns amount $c$ but doesn't get the dollar back for total earnings $c-1$. If the sum is not $7$, he loses his dollar for total earnings $-1$. For this to be a fair game, with expected earnings of $0$, what should be the value of $c$?
A wizard decides to breed newts until she has both a male and a female newt. Newts are born one at a time, each birth is independent and results in a male or a female with probability $\frac{1}{2}$, what is the expected value of the number of newts she will have?
Suppose random variable $X$ has the following pdf: $$f(x) = \begin{cases} x, &0 \le x \le 1\\ 2-x, & 1 < x \le 2\\ 0, &otherwise\end{cases}$$. Compute $E[X]$.
A random variable has $P(X=x)=x/15$ for $x=1, 2, 3, 4, 5$, and $0$ otherwise. Find the expectation and variance of $X$.
Can we have a random variable with $E[X]=3$ and $E[X^2]=8$? Give an example or prove that it is impossible.
Suppose $P (X \in\{1, 2, 3\})=1$ and $E[X]=2.5$. What are the smallest and largest possible values for the variance?
Suppose $X$ and $Y$ are independent with $E[X]=1$, $E[Y]=2$, $V[X]=3$, and $V[Y]=1$. Find the mean and variance of $3X+4Y-5$.
At a local high school, 12 jocks and 4 goths are applying to UC Berkeley.
Suppose that the jocks have a 0.1 chance of getting in and the goths have a 0.2 chance. Assume all admissions decisions are independent. Find the mean and variance of the number of students accepted.
Suppose $X$ takes on the values $−2, −1, 0, 1, 2$ with probability $1/5$ each, and let $Y=X^2 $.
(a) Find $COV [X,Y]$.
(b) Are $X$ and $Y$ independent?
Bakers Ellen and Mina each begin with 1 pound of flour. Ellen chooses a quantity A at random from a uniform distribution on $[0,1]$. Ellen gives $A/2$ to Mina and uses $A/2$ to bake a cake. Mina chooses a quantity B at random from a uniform distribution on $[0,1]$ (A and B are independent). Mina gives $B/2$ to Ellen and uses $B/2$ to bake a cake. Let E be the amount of flour Ellen is left with at the end and let M be the amount of flour Mina is left with at the end. Compute $cov[E,M]$