The continuous random variables $X$ and $Y$ have the joint probability density function:
$$f(x, y)= \begin{cases}
\dfrac{3}{2}y^2, & \text{ where } 0\leq x \leq 2 \text{ and } 0 \leq y \leq 1 \\[2ex]
0, & \text{ otherwise} \\
\end{cases}
$$
I am asked to find the marginal distributions of $X$ and $Y$, and show that $X$ and $Y$ are independent.
I know the marginal distribution to be the probability distribution of a subset of values, does that mean the marginal distribution can be obtained by calculating the probability distribution of the piecewise function in locations where $f(x, y)$ does not equal zero?
Best Answer
Yes. In this case, the subsets of $\{X, Y\}$ we're interested in are $\{X\}$ and $\{Y\}$.
You have been given the joint density function, $f_{X,Y}(x,y)$, and the support for this function, $0\leq x\leq 2, 0\leq y\leq 1$. To obtain the marginal density functions, you integrate over the support for the unwanted variable.
$$\begin{align} f_X(x) & = \int_{0}^1 f_{X,Y}(x,\color{blue}{y})\operatorname d \color{blue}{y} & : \big[0\leq x\leq 2\big] \\[1ex] & = \tfrac 3 2 \int_0^1 \color{blue}y^2 \operatorname d \color{blue}y \\[1ex] & = \tfrac 1 3 \\[2ex] f_Y(y) & = \int_0^2 f_{X,Y}(\color{blue}{x}, y)\operatorname d \color{blue}{x} & : \big[0\leq y\leq 1\big] \\[1ex] & = \tfrac 3 2 y^2 \int_0^2 \operatorname d \color{blue}x \\[1ex] & = 3 y^2 \end{align}$$