(See Steps) Let X and Y have joint probability density function given by f(x, y)=k x^2(1-y)^2, 0 ≤q x ≤q 1, 0 ≤q y ≤q 1 Find each of the following:

Question: Let $X$ and $Y$ have joint probability density function given by

\[f(x, y)=k x^{2}(1-y)^{2}, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1\]

Find each of the following: (a) normalising constant $k$, (b) marginal distributions of $X$ and of $Y$, (c) conditional distribution of $X$ given $Y$, and hence the conditional expectation and variance of $X$ given $Y$, (d) covariance and correlation between $X$ and $Y$. Are the random variables independent?

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(See Steps) Let X and Y have joint probability density function given by f(x, y)=k x^2(1-y)^2, 0 ≤q x ≤q 1, 0 ≤q y ≤q 1 Find each of the following:

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