Computing marginal probability density function – integration question

definite integralsintegrationmarginal-distributionmultivariable-calculusprobability

Just stuck trying to evaluate a certain integral for the following question.

Let $(X,Y)$ be a bivariate random variable with joint probability density function:
$$
f_{X,Y}(x,y)
= \begin{cases}
\frac{6y}{x^2} &\text{for }\quad 0<x<1,\quad 0<y<x^2\\
0, &\text{elsewhere}
\end{cases}
$$
Compute the marginal probability density functions for $X$ and $Y$.

I'm able to compute the marginal pdf for $X$ but when I try to compute the marginal pdf for $Y$ using the following: $$f_Y(y) = \int_{-\infty}^{\infty}f_{X,Y}(x,y)\:dx=\int_{0}^{1}\frac{6y}{x^2}\:dx$$
I find that the rightmost integral diverges…is this, in fact, true or is there some conceptual misunderstanding on my part here?

Edit: fixed typo in the pdf

Best Answer

HINT

  1. Your pdf does not integrate to $1$, you need to rescale or check for typos.
  2. There are 2 way to parameterize your region $D$: for integrating $dy$ first, you have $0 < x < 1, 0 < y < x^2$. For integrating $dx$ first, $0 < y < 1, \sqrt{y} < x < 1$.

Can you finish?

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