3. Suppose X,,X2,, is a random sample from a standard normal distribution and let Z be...
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
Let X1, X2, ..., Xn be a random sample of size 5 from a normal population with mean 0 and variance 1. Let X6 be another independent observation from the same population. What is the distribution of these random variables? i) 3X5 – X6+1 ii) W, = - X? iii) Uz = _1(X; - X5)2 iv) Wą +xz v) U. + x vi) V5Xe vii) 2X
Let Yi, Ys,.., Y's be a random sample of size 5 from a normal distribution mean 0 and standard deviation 1 and let-3x /5 . Let Y6 be another independent observation from the same distribution. Find the distributions of the following random variables i-1 2(572 +Y) (b) WW
Let Yi, Ys,.., Y's be a random sample of size 5 from a normal distribution mean 0 and standard deviation 1 and let-3x /5 . Let Y6 be another independent observation from...
Let X be a standard normal distribution. Let ξ be another random variable, independent of X, which can take only two possible values, say -1 and 1. Moreover, assume that Ele] = 0. ( . (b) Find COV(x,Y). (c) Are X and Y independent? (d) Is the pair (X,Y) bivariate normal? a) Find the distribution of Y -£X
Let X be a standard normal distribution. Let ξ be another random variable, independent of X, which can take only two possible...
(2) Let X, X, be a random sample from normal distribution N (,o2), stribution N(u, a and let S2 be the sample variance: (a) [8pts] show that ES-g? (b) [8pts] For a random sample of size 2 (i.e. n 2), derive that /02 ~ Z2 where Z has standard normal distribution.
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) PIX Y> Z+2 (b) Var3X+4Y
f a random sample X,X, X, from the 2. Let Y, < Y.< Y, be the order statistics o exponential distribution with mean β. Let (i) Are the random variables U,V,W independent? (ii) What is the distribution of each of U,V and W.
9.28 Let X and Y be independent standard normal random variables. Find the mgf of X2 +Y2. What can you conclude about the distribution of X2 +Y2? (Hint: See Example 9.19.)