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- Suppose X1, X2,.., Xn are independent random variables and each of them has a PMF Px(x) k forx = 1,2, 3, 4, 5 What is the value of k? What are the expected value and variance for each X;? What are the expected value and variance for X = E X;?Suppose the variance of the difference of two random variables X and Y is zero. Then, show that X = Y + c for some constant c almost surely. Also find the value of c.Suppose that X1, X2, X3 and X4 are all independent random variables, each with a normal distribution with mean 9 and variance 25. Define X-bar = (X1 + X2 + X3 + X4) / 4. Find Prob(X-bar > 10).
- Let Y1, Y2, Y3, Y4, and Y, be i.i.d. random variables from a population with mean µ and variance o². Further, let Y = (Y, +Y½ +Y3 +Y4+Y;) denote the average of these five random variables. Now consider a different estimator of µ called W w = }Y + }Y½ + }Y, + }Y, + Y, What is the variance of W? That is, calculate Var|W] Hint: Use the properties of expectations, variances, etc. 187 800 800 187An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.
