839 B
839 B
Joint Distributions
Source: probabilitycourse.com
Joint distributions are multivariate probability distributions.
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
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)}