Conditional and Marginal Probability Functions

The marginal and conditional probability functions each describe a single component of a joint distribution: the marginal PMF $p_X(x)$ is the distribution of $X$ on its own, and the conditional PMF $p_{X \mid Y}(x \mid y)$ is the distribution of $X$ once $Y = y$ is known.

$p_X(x) = \sum_y p_{X,Y}(x, y)$

$p_{X \mid Y}(x \mid y) = \frac{p_{X,Y}(x, y)}{p_Y(y)}$

The marginal is found by summing (or integrating, in the continuous case) the joint PMF over the other variable.
The conditional is defined wherever $p_Y(y) > 0$ ; it renormalizes the joint by the marginal of the conditioning variable.
If $X$ and $Y$ are independent, the conditional equals the marginal: $p_{X \mid Y}(x \mid y) = p_X(x)$ .

Insurance Claims by Policy Type Let $X$ = number of claims (0 or 1) and $Y$ = policy type (1 or 2), with joint PMF $p(0,1)=0.3$ , $p(1,1)=0.2$ , $p(0,2)=0.1$ , $p(1,2)=0.4$ . Find the marginal distribution of $X$ and $P(X=1 \mid Y=2)$ .

Answer Marginals of $X$ : $p_X(0) = 0.3 + 0.1 = 0.4$ and $p_X(1) = 0.2 + 0.4 = 0.6$ . Marginal of $Y$ at 2: $p_Y(2) = 0.1 + 0.4 = 0.5$ . The conditional is: $p_{X \mid Y}(1 \mid 2) = \frac{p(1,2)}{p_Y(2)} = \frac{0.4}{0.5} = 0.8$ Given a type-2 policy, there is an 80% chance of a claim.