(a) 0.5 pt - Write two ways to compute the variance of X using the expectation operator (b) 0.5 p...
Q2. More about operations with expectation and covariances Recall that the variance of random variable X is defined as Var(X) Ξ E 1(X-E(X))2」, the covariance is Cor(X, Y-E (X-E(X))(Y-E(Y)), and the correlation is Corr(X,Y) Ξ (a) What is the value of EX-E(X))? (Hint: Let μ denote E(X). Then, the parameter μ is a unknown, but fixed value like a constant.) (0.5 pt) b) The following is the proof that Var(X) E(X2) E(X)2: -E(x)-E(x)2 In a similar way, prove that Cov(X,...
1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is 0.5 x=1 0.5 x=-1 Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y are independent. Meanwhile, let Z = XY. N(0,1). X and Y (a) What is the Expectation (mean value) of X? 3pts (b) Are Y and Z independent? (Just clarify, do not need to prove) [2pts c)...
1. Two independent random variables X and y are given with their distribution laws 4 P 07 0.1 0.2 P 0.2 0.3 0.5 Find 1) the variance of random variable Y 2) the distribution law of random variable Z-0.5Y+x END TEST IN PROBABL ITY THEORY AND STAISTICS Variant 1 1. Two independent random vanables X and Y are given with their distribution laws: 2 0.7 0.1 P 0.2 0.3 0.5 0.2 Find 1) the variance of random varñable Y 2)...
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B) (a) If var[X o2 for each Xi (i...
Question 7 6 pts . If a continuous randaom variable X has a bell curve (normal distribution with mean 5 and variance 25, using the z table I gave you, find P(X < 5) Note X is continuous, X<5 same as X < =5 ) If a discrete randaom variable X has a approximate bell curve (normal) . distribution with mean 5 and variance 25, using the z table I gave (Here X<5 is not the same as X you,...
Write solutions legibly, and show all work. Walk the reader through your thought process, using English words when necessary. 1. Recall question 2 of the previous homework – We draw 6 cards from a 52 card deck and let X = the number of heart cards drawn. You already found the pmf back then. You’re allowed to use it here without re-deriving it. a. What is the expected value of X? b. What is the variance of X? What is...