H/W: if X and Y are continuous random variables, prove E[X] = E[E[X|Y]]? %3D
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- Q1 Let X1 and X2 be independent exponential random variables with identical parameter A. Q1(i.) Find the distribution of Z = max(X1, X2). Q1 (ii.) Find the distribution of Y = min(X1, X2). Q1(iii.) Calculate E[Y]. Q1(iv.) Calculate E[Z]. Q1(v.) Using the relation Z = X1+X2 – Y, Calculate E[Z] and verify that it agrees with the calculation done in part (iv.)The PDF of a random variable Y is Sy/2 0If X and Y are Gaussian random variables then what is E[XY]?Prove the theoremProve the propertyLet X be a continuous random variable with cdf F(x). Show that E(I(X < x)) = F(r) where I is the indicator function (1 if XWhich of the following is false for continuous and independent random variables Y and Z?Let Q be a continuous random variable with PDF | 6q(1 – q) if 0 < q < 1 fo(q) = otherwise This Q represents the probability of success of a Bernoulli random variable X, i.e., P(X =1|Q = q) = q. Find foix (q|x) for x E {0, 1} and all q.(Revision.) Let X = Wo.5 + 0.5W1 – 2W2 – W3, where (W1, t > 0) is standard BM. Find the expectation E(X²).Let X be a random variable with CDF x > 1 Fx(x) = 0 < x < 1 %3D x < 0 a. What kind of random variable is X: discrete, continuous, or mixed? b. Find the PDF of X, fx(x). c. Find E(ex).Let A and B be independent exponential random variables, both with mean 1. If U = A + B and V = A/B, find the joint pdf of U and V.Let X be a continuous random variable with p.d.f. f(x) and distribution function F(x). If Y = X², (a) what is the g(y)? 14/12 (b) If ƒ(x) = -√2/² 2π -∞Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON