Let Y₁ and Y₂ be independent normal random variables, each with mean 0 and variance o². Define U₁ = Y₁ + Y₂ and U₂ = Y₁ - Y₂. Show that U₁ and U₂ are independent normal random variables, each with mean 0 and variance 20². [Hint: If (U₁, U₂) has a joint moment-generating function m (1₁, 12), then U₁ and U₂ are independent if and only if m(1₁, 12) = mʊ₁ (1₁)mʊ₂ (1₂).]

Identify the distributions of the random variables with the following moment- generating functions and derive the mean and the variance: a) My(t) = (1 – 4t)-2 b) My(t) = e-5t+6t² %3D

Let U, V be two uniform independent random variables on [0, 2]. (a) Given U = 1, find the conditional expectation of 3U + 4V. (b) Given U = 1, find the conditional expectation of 3eU+V. (c) Given U = 1, find the conditional variance of 3U + 4V. (d) Given U + V = 3, find the conditional expectation of U - V and 3U + 4V. (Hint: consider the map g(u, v) = (u - v, u + v).)

We have the following information about the random variables X and Y: 0.5, дY – 1,o = 1, o = 2.25. Calculate the variance of Z 1X + 7Y, | a) when the coefficient of correlation is p(X, Y) = 0.33: b) when X and Y are independent random variables: