Let $\displaystyle X \;and \; Y $be random variables. Suppose that the conditional distribution of $\displaystyle X \; given \; Y=y$ is binomial distribution with parameters $\displaystyle n \;and \; y.$ Assume that Y is a $\displaystyle continuous \; uniform\; (0,1) $distribution. Find the marginal distribution of X.

I think that:

since, $\displaystyle P(X|Y) = \dfrac{f(x,y)}{f_{Y}(y)}$

but $\displaystyle f_{Y}(y) = 1$

so, $\displaystyle f(x,y) = \text{Binomial}(n,y)$

is this correct? I am confused on finding the marginal pdf. Can anyone help?