Joint PDFs and Multiple Random Variables

Joint PDF

For two continuous random variables $X$ and $Y$, the joint PDF $f_{X,Y}(x, y)$ describes their combined distribution:

$$P((X, Y) \in A) = \iint_A f_{X,Y}(x, y) \, dx \, dy$$

Properties

• $f_{X,Y}(x, y) \geq 0$ for all $x, y$
• $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx \, dy = 1$

Marginal PDF

From the joint PDF, find the marginal PDF of each variable:

$$f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dy$$

$$f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx$$

We “integrate out” the other variable.

Conditional PDF

The conditional PDF of $X$ given $Y = y$:

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

Interpretation

This describes the distribution of $X$ when we know $Y = y$. It’s a “slice” of the joint PDF, normalized.

Independence

$X$ and $Y$ are independent if and only if:

$$f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y) \quad \text{for all } x, y$$

Equivalently: $f_{X|Y}(x|y) = f_X(x)$ (knowing $Y$ doesn’t change the distribution of $X$).

Expected Values

Joint Expectation

For a function $g(X, Y)$:

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

Linearity

$$E[X + Y] = E[X] + E[Y]$$

Always holds, even for dependent variables.

Product (Independent Case)

If $X$ and $Y$ are independent:

$$E[XY] = E[X] \cdot E[Y]$$

Derived Distributions

If $Y = g(X)$, how do we find the PDF of $Y$?

Method 1: CDF Approach

1. Find $F_Y(y) = P(Y \leq y) = P(g(X) \leq y)$
2. Differentiate: $f_Y(y) = \frac{d}{dy} F_Y(y)$

Example: $Y = aX + b$

If $X$ has PDF $f_X(x)$ and $Y = aX + b$ with $a > 0$:

$$f_Y(y) = \frac{1}{a} f_X\left(\frac{y - b}{a}\right)$$

Convolution

For $Z = X + Y$ where $X$ and $Y$ are independent:

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

This is the convolution of the two PDFs.

Example: Sum of Two Uniforms

If $X, Y \sim \text{Uniform}(0, 1)$ and independent, then $Z = X + Y$ has a triangular PDF on $[0, 2]$.