Consider the stochastic process W* = {W, :t≥ 0}, where W√Z, Z-N(0, 1). (a) (2 marks). Using the results in Chapter 3, find an approximate 95% confi- dence interval for the random variable W,. (b) (3 marks). Determine whether or not W* is a Wiener process.
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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.B) Let the random variable X have the moment generating function e3t M(t) for -1b) A random variable X follows an exponential distribution, X~Exp(0), with parameter 0 > 0. Find the cumulative distribution function (CDF) of X. i) ii) Show that the moment generating function (MGF) of X is M(t) = iii) Use the MGF in (ii) to find the mean and variance of X.Suppose that Z₁, Z2, ..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y=)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z, i = 1, ..., .., n. (b) Determine My(t) for the special case that Z₁ ~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.(b)Let X be a random variable with p.d.f, f(x)=c x=1,2,3,., zero elsewhere. Find the constant c.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 ~ N(0,o²) and Y ~ N(0, v²) be independent random variables and Z = X +Y. (a) Find the LMMSE estimator of X given Z. (b) Find the conditional distribution fzix(z, x).Let P be a random variable having a uniform distribution with minimum 0 and maximum 3 i.e. P ~ Uniform(0, 3). Let Q = log (P/(3-P)). Find E[P]. You areexpected to solve this problem without using Method of Transformation.The potholes in a long road occur as a Poisson process of rate 4 per mile. In other words, if we let R(t) = the number of potholes in the first t miles of the road then the random variables (R(t): t > 0) form a Poisson process of rate 4.
