1. Let $p(x,y)$, $x=1,2,3$ and $y=1,2$, be the joint frequency function for two random variables $X$ and $Y$. Suppose that $p(1,1) = 2\,c$, $p(2,1) = c$, $p(3,1) = c$, $p(1,2) = 2\,c$, $p(2,2) = 3\,c$, $p(3,2) = 3\,c$. Find the constant $c$, and calculate $E[X]$.

    $E[X] = {{17}\over{12}}$ $E[X] = {{41}\over{12}}$ $E[X] = {{7}\over{12}}$ $E[X] = 2$

  2. Let $X$ and $Y$ be independent random variables. Suppose that $Var(X) = 2$, $Var(Y) = 4$. Then find the variance for Z = −3Y − 2X.

    $Var(Z) = -16$ $Var(Z) = -30$ $Var(Z) = 44$ $Var(Z) = 6$

  3. Let $p(k)$, $k=-1,0,1$, be the frequency function for random variable $X$. Suppose that $p(0) = {{2}\over{5}}$, and that $p(-1)$ and $p(1)$ are unknown. Find $E[X^2]$.

    $\displaystyle {{2}\over{5}}$ $\displaystyle {{3}\over{5}}$ $\displaystyle {{1}\over{4}}$ $\displaystyle {{1}\over{5}}$

  4. Suppose that $\displaystyle {{1}\over{3}}$ of college students binge drink. Let $X$ be the number of students who binge drink out of sample size $n= 10$. Find the mean and variance of $X$.

    $E[X] = {{20}\over{9}}$ and $Var(X) = {{10}\over{3}}$ $E[X] = {{36}\over{25}}$ and $Var(X) = {{9}\over{5}}$ $E[X] = {{9}\over{5}}$ and $Var(X) = {{36}\over{25}}$ $E[X] = {{10}\over{3}}$ and $Var(X) = {{20}\over{9}}$

  5. A pair $(X,Y)$ of discrete random variables has the joint frequency function $p(x,y) = \frac{xy}{18}$, $x=1,2,3$ and $y=1,2$. Find $P(X+Y= 4)$.

    $\displaystyle {{2}\over{9}}$ $\displaystyle {{7}\over{18}}$ $\displaystyle {{1}\over{18}}$ $\displaystyle {{1}\over{3}}$

  6. Determine the constant $c$ so that $p(x)$ is a frequency function if $p(x) = c\left( {{1}\over{2}}\right)^x$, $x=1,2,3,\ldots$.

    1 2 2 $\displaystyle {{1}\over{2}}$

  7. Let $p(x)$, $x=-1,0,1$, be the frequency function for random variable $X$. Suppose that $p(0) = {{1}\over{3}}$, and that $E[X] = -{{1}\over{6}}$. Find $p(-1)$ and $p(1)$.

    $p(-1) = {{1}\over{4}}$ and $p(1) = {{5}\over{12}}$ $p(-1) = {{5}\over{12}}$ and $p(1) = {{1}\over{4}}$ $p(-1) = {{1}\over{30}}$ and $p(1) = {{1}\over{5}}$ $p(-1) = {{1}\over{5}}$ and $p(1) = {{1}\over{30}}$

  8. Suppose that $E[X] = -1$, $E[Y] = -3$. Then find the mean for Z = 3Y + 2X.

    $E[Z] = -4$ $E[Z] = 5$ $E[Z] = -9$ $E[Z] = -11$

  9. Suppose that $\displaystyle {{1}\over{4}}$ of college students binge drink. Let $X$ be the number of students who binge drink out of sample size $n= 6$. Find $P(X= 0)$.

    $\binom{ 6}{ 0} {{64}\over{729}} = {{64}\over{729}}$ $\binom{ 6}{ 0} {{729}\over{4096}} = {{729}\over{4096}}$ $\binom{ 6}{ 0} {{1}\over{4096}} = {{729}\over{4096}}$ $\binom{ 6}{ 0} {{1}\over{729}} = {{64}\over{729}}$

  10. Let $p(x,y)$, $x=1,2,3$ and $y=1,2$, be the joint frequency function for two random variables $X$ and $Y$. Suppose that $p(1,1) = 0$, $p(2,1) = 0$, $p(3,1) = 0$, $p(1,2) = 0$, $p(2,2) = {{1}\over{3}}$, $p(3,2) = {{2}\over{3}}$. Are $X$ and $Y$ independent?

    $X$ and $Y$ are independent . $X$ and $Y$ are not independent .


Department of Mathematics
Last modified: 2025-05-26