2018.09.11

$\DeclareMathOperator{\Var}{Var}$ $\DeclareMathOperator{\Cov}{Cov}$ $\DeclareMathOperator{\Span}{Span}$

  1. Suppose $X, Y$ and $\epsilon$ are random variables.

    • $X$ is chosen uniformly at random from the integers ${1,2,\ldots,10}$
    • $Y = X + \epsilon$.
    • $X$ and $\epsilon$ are independent.
    • The standard deviation of $Y$ is 4.

    Find the standard deviation of $\epsilon$.

  2. Using the table below, find a bunch of stats (below).

    x y=4 y=8
    x=2 0.1 0.2
    x=6 0.1 0.3
    x=10 0.2 0.1
    1. $P(X=6 | Y=8)$
    2. $P(X=10)$
    3. $P(Y=8)$
    4. $E[X]$, $E[Y]$
    5. $\Var(X)$
    6. $\Cov(X,Y)$

  3. Given the following information:

    E[X] E[Y] E[XY] E[X^2] E[Y^2]
    2 3 8 5 12

    Find $\Var(X)$, $\Var(Y)$, $\Cov(X,Y)$.

  4. If $X \ge 0$, is it true that $E[X^2] = E[X]^2$? Prove or give a counterexample.

  5. Suppose that there are $n$ random variables, $X_i$, all with mean $\mu$ and variance $\sigma^2$. Also $\Cov(X_i,X_j) = 0.3 \sigma^2$.

    Find the variance of the average of the $X_i$ in terms of $\sigma$.