1. 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 $P(X+Y= 5)$.

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

  2. Determine the constant $c$ so that $p(x)$ is a frequency function if $p(x) = cx$, $x=1,2,\ldots, 9$.

    $\displaystyle {{1}\over{15}}$ $\displaystyle {{1}\over{90}}$ $\displaystyle {{1}\over{30}}$ $\displaystyle {{1}\over{45}}$

  3. Let $X$ and $Y$ be independent random variables. Suppose that $Var(X) = 5$, $Var(Y) = 1$. Then find the variance for Z = 3X − 3Y.

    $Var(Z) = 54$ $Var(Z) = 12$ $Var(Z) = 0$ $Var(Z) = 6$

  4. 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 $P(X= 0)$.

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

  5. 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 $n= 9$. Find the mean and variance of $X$.

    $E[X] = 3$ and $Var(X) = 2$ $E[X] = {{5}\over{2}}$ and $Var(X) = {{15}\over{8}}$ $E[X] = 2$ and $Var(X) = 3$ $E[X] = {{15}\over{8}}$ and $Var(X) = {{5}\over{2}}$

  6. Determine the constant $c$ so that $p(x)$ is a frequency function if $p(x) = c\left( {{3}\over{4}}\right)^x$, $x=1,2,3,\ldots$.

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

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

    $E[Z] = 0$ $E[Z] = 1$ $E[Z] = -6$ $E[Z] = 6$

  8. Let $p(x)$, $x=-1,0,1$, be the frequency function for random variable $X$. Suppose that $p(0) = {{1}\over{4}}$, and that $E[X] = -{{7}\over{20}}$. Find $p(-1)$ and $p(1)$.

    $p(-1) = {{1}\over{20}}$ and $p(1) = {{2}\over{5}}$ $p(-1) = {{2}\over{5}}$ and $p(1) = {{1}\over{20}}$ $p(-1) = {{11}\over{20}}$ and $p(1) = {{1}\over{5}}$ $p(-1) = {{1}\over{5}}$ and $p(1) = {{11}\over{20}}$

  9. Let $p(k)$, $k=-1,0,1$, be the frequency function for random variable $X$. Suppose that $p(0) = {{1}\over{4}}$, and that $p(-1)$ and $p(1)$ are unknown. Find $E[X^2]$.

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

  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 $p(1,1) = 2\,c$, $p(2,1) = 3\,c$, $p(3,1) = 2\,c$, $p(1,2) = 2\,c$, $p(2,2) = c$, $p(3,2) = 2\,c$. Find the constant $c$, and calculate $E[XY]$.

    $E[XY] = {{17}\over{6}}$ $E[XY] = {{7}\over{6}}$ $E[XY] = {{5}\over{3}}$ $E[XY] = 34$


Department of Mathematics
Last modified: 2025-04-08