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- Please state your assumptions and show all your work clearly. 1. Show that the following equalities hold: a) Population variance: N 'i=1 i=1 N N b) Sample variance: E,(xi – x)² E-,(x;)² n(x)? i=1 n – 1 n – 1 n – 1 c) Sum of differences from the population mean: Σα. > (xi – u) = 0 | i=14.1. Suppose X and Y are independent random variables, which are jointly continuous. x' is equal to the distribution 1. Show carefully that the distribution of 'X + Y given X of the random variable à +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). = 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].1. Random variable X has p.d.f -6z 6e fx(z) otherwise and let Y = eX. Calculate E(X)
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