# Joint Distributions

**Source:** probabilitycourse.com

Joint distributions are **multivariate probability distributions**.

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## Conditional Probability

$$P(A|B) = \frac{P(A \cap B)}{P(B)}, \text{ when } P(B) > 0$$

### Example: Fair Die

If this is a fair die, what's the PMF of the outcomes given the event $A = \{x < 5\}$?

$$P(A) = \frac{4}{6}$$

$$P_{X|A}(1) = \frac{P(X = 1 \cap x < 5)}{P(x < 5)} = \frac{\frac{1}{6}}{\frac{4}{6}} = \frac{1}{4}$$

$$P_{X|A}(2) = P_{X|A}(3) = P_{X|A}(4) = P_{X|A}(1) = \frac{1}{4}$$

$$P_{X|A}(5) = \frac{P(x = 5 \cap x < 5)}{P(x < 5)} = \frac{0}{\frac{4}{6}} = 0$$

$$P_{X|A}(6) = 0$$

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## Two Random Variables

When working with two random variables:

$$P_{X|Y}(x_i | y_j) = P(X = x_i | Y = y_j)$$

$$= \frac{P(X = x_i \cap Y = y_j)}{P_Y(y_j)}$$

$$= \frac{P_{XY}(x_i, y_j)}{P_Y(y_j)}$$