Joint Distributions and Independence

Definition. The joint CDF of $(X, Y)$ is $F_{X,Y}(x, y) = P(X \leq x, Y \leq y)$.

Definition. The joint PDF (for continuous random variables) is $f_{X,Y}(x, y) \geq 0$ such that

$F_{X,Y}(x, y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f_{X,Y}(u, v)\, du\, dv$

Definition. The marginal PDF of $X$ is $f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y)\, dy$.

Definition. The covariance of $X$ and $Y$ is

$\mathrm{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]$

Proposition 2.6. $\mathrm{Cov}(X, Y) = \mathrm{Cov}(Y, X)$ and $\mathrm{Cov}(aX + b, cY + d) = ac\,\mathrm{Cov}(X, Y)$.

Definition. The correlation coefficient is

$\rho(X, Y) = \frac{\mathrm{Cov}(X, Y)}{\sqrt{\mathrm{Var}(X)\,\mathrm{Var}(Y)}}$

Theorem 2.7 (Cauchy—Schwarz for Random Variables). $|\rho(X, Y)| \leq 1$, with equality if and only if $Y = aX + b$ almost surely for some $a, b$.

Definition. $X$ and $Y$ are independent if $F_{X,Y}(x, y) = F_X(x)\, F_Y(y)$ for all $x, y$.

For continuous random variables, this is equivalent to $f_{X,Y}(x, y) = f_X(x)\, f_Y(y)$.

Proposition 2.8. If $X$ and $Y$ are independent, then $\mathrm{Cov}(X, Y) = 0$. The converse is false.

Definition. The conditional PDF of $Y$ given $X = x$ is:

$f_{Y|X}(y|x) = \frac{f_{X,Y}(x, y)}{f_X(x)}$

provided $f_X(x) > 0$.

Definition. The conditional expectation of $Y$ given $X = x$ is:

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

The random variable $E[Y | X]$ has the property that $E[E[Y|X]] = E[Y]$ (law of total expectation).

The bivariate normal distribution has joint PDF:

$f_{X,Y}(x, y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1 - \rho^2}} \exp\left(-\frac{1}{2(1 - \rho^2)}\left[\frac{(x - \mu_X)^2}{\sigma_X^2} - 2\rho\frac{(x - \mu_X)(y - \mu_Y)}{\sigma_X\sigma_Y} + \frac{(y - \mu_Y)^2}{\sigma_Y^2}\right]\right)$

where $\rho = \mathrm{Corr}(X, Y)$.

Properties of the bivariate normal:

• Marginal distributions are normal: $X \sim N(\mu_X, \sigma_X^2)$, $Y \sim N(\mu_Y, \sigma_Y^2)$.
• Conditional distribution is normal: $Y | X = x \sim N\left(\mu_Y + \rho \frac{\sigma_Y}{\sigma_X}(x - \mu_X),\ \sigma_Y^2(1 - \rho^2)\right)$.
• $X$ and $Y$ are independent if and only if $\rho = 0$.

For a transformation $(U, V) = g(X, Y)$ where $g$ is a diffeomorphism, the joint PDF is:

$f_{U,V}(u, v) = f_{X,Y}(g^{-1}(u, v)) \cdot |\det J_{g^{-1}}|$

where $J$ is the Jacobian matrix of the inverse transformation.

Example. Let $X, Y$ be independent exponential($\lambda$) random variables. The joint PDF of $U = X + Y$ and $V = X/(X + Y)$ factors as $f_{U,V}(u, v) = \lambda^2 u e^{-\lambda u} \cdot \chi_{(0,1)}(v)$, showing $U$ and $V$ are independent with $U \sim \mathrm{Gamma}(2, \lambda)$ and $V \sim \mathrm{Uniform}(0, 1)$.

If $X$ and $Y$ are independent, the PDF of $Z = X + Y$ is the convolution:

$f_Z(z) = \int_{-\infty}^{\infty} f_X(z - y)\, f_Y(y)\, dy = \int_{-\infty}^{\infty} f_X(x)\, f_Y(z - x)\, dx$

For moment generating functions: $M_Z(t) = M_X(t)\, M_Y(t)$.

Problem 1. Let $X$ and $Y$ have joint PDF $f_{X,Y}(x, y) = c(1 - x^2 - y^2)$ for $x^2 + y^2 \leq 1$. Find $c$, the marginal PDFs, and $P(X > 0, Y > 0)$.

Problem 2. Show that if $\mathrm{Cov}(X, Y) = 0$ for a bivariate normal pair, then $X$ and $Y$ are independent.

Problem 3. Let $X$ and $Y$ be independent standard normal variables. Find the distribution of $R = \sqrt{X^2 + Y^2}$ and $\Theta = \arctan(Y/X)$.

Solution. $R$ has Rayleigh distribution with PDF $f_R(r) = r e^{-r^2/2}$ for $r \geq 0$, and $\Theta \sim \mathrm{Uniform}(0, 2\pi)$, and $R$ and $\Theta$ are independent. $\blacksquare$

Problem 4. Suppose $X \sim N(0, 1)$ and $Y = X + Z$ where $Z \sim N(0, 1)$ is independent of $X$. Find the joint distribution of $(X, Y)$ and compute $\mathrm{Cov}(X, Y)$.

Law of total probability: $f_Y(y) = \int_{-\infty}^{\infty} f_{Y|X}(y|x)\, f_X(x)\, dx$.

Bayes’ theorem: $f_{X|Y}(x|y) = \frac{f_{Y|X}(y|x)\, f_X(x)}{f_Y(y)} = \frac{f_{Y|X}(y|x)\, f_X(x)}{\int f_{Y|X}(y|t)\, f_X(t)\, dt}$.

The joint moment generating function of $(X, Y)$ is:

$M_{X,Y}(s, t) = E[e^{sX + tY}]$

If $X$ and $Y$ are independent, then $M_{X,Y}(s, t) = M_X(s)\, M_Y(t)$. Joint moments can be recovered by differentiation:

$E[X^m Y^n] = \frac{\partial^{m+n}}{\partial s^m \partial t^n} M_{X,Y}(s, t)\big|_{s=t=0}$

Problem 5. Let $(X, Y)$ have joint PDF $f_{X,Y}(x, y) = 6x$ for $0 < x < 1$, $0 < y < 1 - x$. Find the marginal PDFs and determine if $X$ and $Y$ are independent.

Problem 6. Suppose $X$ and $Y$ are independent exponential random variables with rate $\lambda$. Find the distribution of $X + Y$ and $X/(X + Y)$. Are they independent?

Problem 7. Show that $\mathrm{Var}(X) = E[\mathrm{Var}(X|Y)] + \mathrm{Var}(E[X|Y])$ (the variance decomposition formula).