Let Z1 and Z2 be independent standard normal random variables. Show that X, Y has a bivariate normal distribution when X = Z1, Y = Z1 + Z2.
Answer to relevant QuestionsDerive the distribution of the range of a sample of size 2 from a distribution having density function f(x) = 2x, 0 < x < 1. If X1 and X2 are independent exponential random variables, each having parameter λ, find the joint density function of Y1 = X1 + X2 and Y2 = eX1. Show that the jointly continuous (discrete) random variables X1, . . . ,Xn are independent if and only if their joint probability density (mass) function f (x1, . . . , xn) can be written as for nonnegative functions gi(x), ...Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter λ. If each event is classified as a type i event with probability pi, i = 1, . . . , n, ∑ pi = 1, ...Verify Equation (6.6), which gives the joint density of X(i) and X(j).
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