Marginal distribution

Definition

Let $\mu$ be a probability distribution on $\mathbb{R}^2$, for example the joint distribution of real random variables $X$ and $Y$. The first marginal distribution of $\mu$ is the distribution $\mu_1$ on $\mathbb{R}$ with $\mu_1(A) = \mu(A \times \mathbb{R})$ for each Borel set $A \subseteq \mathbb{R}$. The second marginal is $\mu_2(B) = \mu(\mathbb{R} \times B)$. If $\mu$ is the law of $(X, Y)$, then $\mu_1$ is the law of $X$ and $\mu_2$ is the law of $Y$.

Remarks

The same definition works on a product $E_1 \times E_2$ of measurable spaces, and for $n$ variables with the projections onto some of the coordinates. In the discrete case, $P(X = x) = \sum_{y} P(X = x, Y = y)$. In the case of a joint density $f(x, y)$, $X$ has the density $f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy$. The name comes from tables: the sums of the rows and the columns of a joint table are written in its margins.

Explanations

L1 human

To get the marginal distribution of $X$ from a joint table of $X$ and $Y$, add up each row. You lose the information about how $X$ and $Y$ are linked, and you keep only the chances of the values of $X$.