Marginal and Conditional Densities

Probability · Axiom Academy

LESSON Marginal and Conditional Densities Extracting single-variable distributions and conditional relationships The marginal density of X is obtained by integrating out Y from the joint density: This gives us the distribution of X alone, collapsing across all possible values of Y. Similarly, we can find the marginal density of Y by integrating out X. The conditional density of X given Y = y describes the distribution of X when we know Y takes a specific value: This is the joint density normalized by the marginal density of Y. Conditional densities allow us to update our beliefs about X based on knowledge of Y. 3. Relationship and Applications The joint, marginal, and conditional densities are related by: Law of Total Probability: We can recover the marginal from conditional and marginal of the conditioning variable. Independence check: X and Y are independent if and only if f(x|y) = f(x) for all x,y.

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