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## Homework Statement

Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is

**Var[Y]**?

## Homework Equations

E[Y] = np

Var(Y) = np(1-p) for variance of a binomial distribution

Var(Y|X) = E(Y^2|X) − {E(Y|X)^2} for conditional variance of y given x

Var(Y) = E[Var(Y|X)] + Var[E(Y|X)] for the law of total variance

## The Attempt at a Solution

Knows that probability X is a uniform (0,1) random variable, we can calculate E(Y). From there, we should be able to calculate Var(Y) using the relevant equations, I think. Using the equation for variance of a binomial distribution and simply plugging in the values for p that we solved considering the uniform (0,1) distribution of X seems too easy/doesn't appear to be correct. My inclination is to use the law of total variance to solve for Var(Y), but it requires calculating Var(Y|X) as well as E(Y^2|X) ? This is where I get stuck, how to calculate E(Y^2|X) given the information I know about Y and X. Using the law of total variance, I also struggle to see how the equation for variance of a binomial distribution comes into play. Any idea if I am the right track/any advice?

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