Bivariate distribution $p(x, y) = \frac{1}{n^2}$ for $x = 1, \dots , n$ and $y = 1, \dots , n$. Find the marginal distributions or $X$ and $Y$.

Question

Let two discrete random variables $X$ and $Y$ have joint distribution

$$p(x, y) = \frac{1}{n^2}$$

for $x = 1, \dots, n$ and $y = 1,\dots,n$.

How would I go about finding the marginal distributions of $X$ and $Y$?

It seems to me that the distribution is uniform, but I'm not sure how to interpret it. Do I pick an $x$ or a $y$ or... I'm not even sure how to begin!

Answer 1

To find the marginal distribution of $X$ in general is to compute $$P(X = j) = \sum_k P(X=j, Y=k).$$

However, in this case we can recognize that $$P(X = j, Y = k) = \frac{1}{n}\cdot \frac{1}{n} = P(X = j)P(Y = k)$$ which implies that $X$ and $Y$ are independent and uniformly distributed on $\{1,2,\dotsc,n\}$.