i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K.
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- Find the moment generating function of the continuous random variable X∼U (a, b).Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.Let X be a discrete random variable with probability mass function P(X= x) =p(1 −p)^x ; x= 0,1,2,.... Here p∈[0,1]. Calculate the moment generating function (MGF) of X, the mean, and variance of this distribution (using the MGF).