Lecture 7

Joint CDFs

The joint CDF of two random variables $X$ and $Y$ is defined as $F_{X,Y}(x,y) = P(X\leq x, Y\leq y) = \int_{-\infty}^{x}{\int_{-\infty}^{y}{f_{X,Y}(s, t)}dt}ds$

Example

$F_{X,Y}(x,y)= \begin{cases} \frac{xy}{25} \text{ if } 0 \leq x, y \leq 5\\ 0 \text{ otherwise} \end{cases}$

The sample space is the square $(0, 0)$ to $(5, 5)$ .

Find $P(X \leq 4, Y \leq 3)$ . $P(X \leq 4, Y \leq 3) = F_{X,Y}(4,3)=\frac{12}{25}$

The joint PDF $f_{X,Y}(x,y)$ can be derived from the joint CDF by taking the derivatives with respect to both $x$ and $y$ . $f_{X,Y}(x,y) = \frac{\delta^2}{\delta x\delta y}F_{X,Y}(x,y)$

Example: Let the joint CDF be

$F_{X,Y}(x,y) = \begin{cases} \frac{xy}{25} \text{ if } 0 \leq x, y \leq 5\\ 0 \text{ otherwise } \end{cases}$

The PDF will be

$\frac{delta^2}{\delta x \delta y} \cdot \frac{xy}{25} = \frac{1}{25}$

The marginal distributions $f_X(x)$ and $f_Y(y)$ can be calculated as follows: $\begin{align} f_X(x) &= \int_{-\infty}^{\infty}{f_{X,Y}(x,y)}dy\\ f_Y(y) &= \int_{-\infty}^{\infty}{f_{X,Y}(x,y)}dx \end{align}$

The expectation of a function of a joint random variable is:

$E[g(X,Y)] = \int_{-\infty}^\infty{\int_{-\infty}^\infty{g(x,y)f_{X,Y}(x,y)}dx}dy$

Conditional PDF

The conditional PDF of a continuous random variable $X$ , given an event $A$ with $P(A) > 0$ , is defined as a nonnegative function

$P(X \in B|A) = f_{X|A}(x)dx$

For any subset $B$ of the real line,

$f_{X|{X \in A}} = \begin{cases} \frac{f_X(x)}{P(X \in A)} \text { if } x \in A\\ 0 \text{ otherwise } \end{cases}$

Example:

The time $T$ until a new light bulb burns out is an exponential random variable with parameter $\lambda$ . Ariadne turns the light on, leaves the room, and when she returns $t$ time units later, finds the light bulb still on, which corresponds to the event $A = \{T \in T\}$ . Let $X$ be the additional time until the light bulb burns out. What is the conditional CDF of $X$ given event $A$ ?

$\begin{align} P(X > x|A) &= P(T > t + x|t > t)\\ &= \frac{P(T > t + x,T > t)}{P(T>t)}\\ &= \frac{P(T > t+x)}{P(T>t)}\\ &= \frac{e^{-\lambda(t+x)}}{e^{-\lambda t}}\\ &= e^{-\lambda x} \end{align}$

Conditioning

For multiple random variables, if we condition on a positive probability event of the form $C=\{(X,Y) \in A\}$ , we have

$f_{XY|C}(x,y) = \begin{cases} \frac{f_{X,Y}(x,y)}{P(C)} \text{ if } (x,y) \in A\\ 0 \text{ otherwise } \end{cases}$

In this case, the conditional PDF of $X$ given this event can be obtained from the formula

$f_{X|C}(x) = \int_{-\infty}^\infty{f_{XY|C}(x,y)}dy$

The above two formulas allow us to obtain the conditional PDF of a random variable $X$ when the conditioning event is not of the form $\{X \in A\}$ but is instead defined in terms of multiple random variables.

Let $X$ and $Y$ be continuous random variables with joint PDF $f_{X,Y}$ . For any $y$ with $f_Y(y) > 0$ , the conditional PDF of $X$ given that $Y=y$ is defined by $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$

Example:

Ben throws a dart at a circular target of radius $r$ . We assume that he always hits the target, and that all points of impact $(x, y)$ are equally likely so that the joint PDF of the random variables $X$ and $Y$ is uniform. Since the area of the circle is $\pi r^2$ , we have:

$f_{X,Y}(x,y) = \begin{cases} c=\frac{1}{\text{area}} \text{ if } (x, y) \text{ is in the circle}\\ 0 \text{ otherwise} \end{cases}$

In other words, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi r^2} \text{ if } x^2+y^2 \leq r^2\\ 0 \text{ otherwise} \end{cases}$

To calculate the conditional PDF $f_{X|Y}(x|y)$ , first find the marginal PDF $f_Y(y)$ . For $|y| \leq r$ , it is given by

$\begin{align} f_Y(y) &= \int_{-\infty}^\infty{f_{X,Y}(x,y)}dx\\ &= \frac{1}{\pi r^2}\int_{x^2+y^2 \leq r^2}dx\\ &= \frac{1}{\pi r^2} \int_{\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}}dx\\ &= \frac{1}{\pi r^2}\sqrt{r^2-y^2} \text{ if } |y| \leq r \end{align}$

Note that the marginal PDF $f_Y$ is not uniform. The conditional PDF is:

$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} = \frac{\frac{1}{\pi r^2}}{\frac{2}{\pi r^2}\sqrt{r^2-y^2}} \text{ if } x^2+y^2 \leq r^2$

CS394 Lectures

Lecture 7

Joint CDFs

Marginal Distributions

Conditional PDF

Conditioning