(Solved) Consider continuous random variables X and Y with joint probability density function f(x, y)=k x, 0 ≤q y ≤q 1-x, 0 ≤q x ≤q 1 . What

Question: Consider continuous random variables $X$ and $Y$ with joint probability density function

\[f(x, y)=k x, \quad 0 \leq y \leq 1-x, 0 \leq x \leq 1 .\]

What value must $k$ take for this to be a proper probability density function? Find the marginal probability density function of $X$ and the cumulative distribution function of

$Y$. Calculate the conditional distribution of $X$ given $Y$ and the conditional expectation of $X$ given $Y$. Are $X$ and $Y$ independent?

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(Solved) Consider continuous random variables X and Y with joint probability density function f(x, y)=k x, 0 ≤q y ≤q 1-x, 0 ≤q x ≤q 1 . What

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