5. Two-Dimensional Random Variables

When dealing with two variables $X$ and $Y$ simultaneously (e.g., Height and Weight).

5.1 Joint Distribution Function

Definition: $F(x, y) = P(X \le x, Y \le y)$. The probability that $X$ is less than or equal to $x$ AND $Y$ is less than or equal to $y$.

Discrete Case (Joint PMF): $P(X=x_i, Y=y_j) = p_{ij}$.

• Condition: Sum of all $p_{ij}$ must be 1.

Continuous Case (Joint PDF): $f(x, y)$.

• Condition: \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) dx dy = 1.

5.2 Marginal Distribution Function

Concept: The distribution of one variable while ignoring the other. We "marginalize" the other variable out.

Formulas:

• Discrete:
• Marginal PMF of X: $P(X=x) = \sum_{\text{all } y} P(X=x, Y=y)$ (Sum across the rows).
• Marginal PMF of Y: $P(Y=y) = \sum_{\text{all } x} P(X=x, Y=y)$ (Sum down the columns).
• Continuous:
• Marginal PDF of X: $f_X(x) = \int_{-\infty}^{\infty} f(x,y) dy$.

5.3 Conditional Distribution Function

Concept: The distribution of $X$ given that $Y$ has already taken a specific value $y$.

Formulas:

• Discrete:

$$ P(X=x | Y=y) = \frac{P(X=x, Y=y)}{P(Y=y)} $$ (Joint Probability divided by Marginal Probability of the condition).

• Continuous:

$$ f(x|y) = \frac{f(x,y)}{f_Y(y)} $$