Examples

Joint and marginal PMFs - Smaller and bigger roll

Roll two dice, and let $X$ indicate the smaller of the numbers rolled, and let $Y$ indicate the bigger number.

Make a chart showing the PMF. Compute the marginal probabilities, and write them in the margins of the chart.

Solution

Event probabilities by reading PMF table

Here is a joint PMF table:

$$\begin{array}{ccccc}{P}_{Q,G}(q,g)& g=0& g=1& g=2& g=3\\ q=0& 0.06& 0.18& 0.24& 0.12\\ q=1& 0.04& 0.12& 0.16& 0.08\end{array}$$

Using the table, compute the following event probabilities:

(a) $P[Q=0]$ $$ (b) $P[Q=G]$ $$ (c) $P[G>1]$ $$ (d) $P[G>Q]$

Joint and marginal PMFs - Coin flipping

Flip a fair coin four times. Let $X$ measure the number of heads in the first two flips, and let $Y$ measure the total number of heads.

Make a chart showing the PMF. Compute the marginal probabilities, and write them in the margins of the chart.

Marginal and event probability from joint density

Suppose the joint density of $X$ and $Y$ is given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}2x{e}^{x^2-y}& y>x^2,x\in [\mathrm{0,1}]\\ 0& \text{otherwise}\end{array}$$

Find (a) $f_Y(y)$ and (b) $P[Y<3X^2]$.

Solution

(a)

When $y\in [\mathrm{0,1}]$, the $x$ range is $0$ to $\sqrt{y}$:

$$\begin{array}{c}{f}_Y(y)={\int }_{-\infty }^{+\infty }{f}_{X,Y}(x,y)dx\\ \\ \gg \gg {\int }_0^{\sqrt{y}}2x{e}^{x^2-y}dx\gg \gg 1-e^{-y}\text{for }0\le y\le 1\end{array}$$

When $y>1$, the $x$ range is $0$ to $1$:

$$\begin{array}{c}{f}_Y(y)={\int }_{-\infty }^{+\infty }{f}_{X,Y}(x,y)dx\\ \\ \gg \gg {\int }_0^{1}2x{e}^{x^2-y}dx\gg \gg e^{1-y}-e^{-y}\text{for }y>1\end{array}$$

Therefore:

$$f_Y(y)=\{\begin{array}{cc}0& y<0\\ 1-e^{-y}& 0\le y<1\\ (e-1)e^{-y}& 1\le y\end{array}$$

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(b)

Find probability of the event $Y<3X^2$:

$$\begin{array}{c}P[Y<3X^2]={\int }_0^1{\int }_{x^2}^{3x^2}2x{e}^{x^2-y}dydx\\ \\ \gg \gg {\int }_0^{1}2x{e}^{x^2}(e^{-x^2}-e^{-3x^2})dx\\ \\ \gg \gg \frac{1}{2}(1+e^{-2})\end{array}$$

Marginals from joint density

The joint PDF for $X$ and $Y$ is given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}6(x+y^2)/5& 0\le x,y\le 1\\ 0& \text{ otherwise }\end{array}$$

Find $f_X(x)$ and $f_Y(y)$.

Event probability from joint density

The joint PDF for $X$ and $Y$ is given by:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}2e^{-x}{e}^{-2y}& x,y>0\\ 0& \text{else}\end{array}$$

Compute $P[X<Y]$.

Properties of joint CDFs

(a) Show with a drawing that if both $x<x^{\prime }$ and $y<y^{\prime }$, we know:

$$F_{X,Y}(x,y)\le F_{X,Y}(x^{\prime },y^{\prime })$$

(b) Explain why:

• $F_X(x)=F_{X,Y}(x,\infty )$
• $F_Y(y)=F_{X,Y}(\infty ,y)$

(c) Explain why:

• $F_{X,Y}(x,-\infty )=0$
• $F_{X,Y}(-\infty ,y)=0$