2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive constants such that a+b= 1, then (i) Find mgf. (ii) If E(X) = m₁, E(X2) = m2, show that m₂ = m₁(2m₁ + 1).
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- Q3/(a) Let X = b(n,) and E(x)= E(16-7x) find n. (b)Let X be a random variable with p.d.f, ƒ(x)= c 3 x=1,2,3,.., zero elsewhere. Find the constant c.Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the aboveX and Y are two independent random variables with fy (x) eX-u (x) and fy (V) = e. u (y). Find the mean of the random variable Z= X - Y for X> Y for. X< Y
- B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.Show that if X', Y' are the deviations of the random variables X and Y from their respective means, then i) r = 1-E (- ii)r = -1+ 2N dy 2N Deduce that -1srs1.If X, Y are standardized random variables and 1+ 2ab a² +6²' r(aX+bY,bX+aY) = find r(X, Y), the coefficient of correlation between X and Y.
- The data of two groups, each group contains two random variables (X1,Y1 and X2,Y2) and each of 30 students is collected. We would like to study the relation between X1,Y1 and X2,Y2 using the measure denoted by T1 andT2, where T1 = E[(Xi – X1)(Yıi – K)]² / [E(x1i - X,)²E(Yu – ¥,)*] ]. T2 = E[(X2i – X2)(Yi – Y)J² / [,E(X2i – X,)² E(Yzi – ¥)²] ]. The data given below, compare between the groups and interpret your results. 1 XI Y1 X2 Y2 98 44 74 52 95 57 79 59 94 66 78 56 81 64 70 74 84 73 79 77 66 65 72 66 73 65 74 50 72 67 98 62 71 66 93 76 67 76 76 84 70 77 86 55 93 66 98 66 92 43 83 65 77 58 68 59 94 56 73 53 76 70 69 74 93 59 90 50 69 59 70 51 86 66 84 55 98 51 78 48 71 64 75 61 95 74 81 53 84 73 73 46 83 69 80 52 78 58 98 61 72 70 79 68 98 53 98 52 73 54 77 56 81 71 84 51 84 49 82 69Let X and Y be two random variables with Var(X) = 6, Var(Y ) = 3, andthe correlation ρ(X, Y ) = −1/4. Find the value of Var(X − 2Y + 7).