Prove linearity of expectations of one or two continuous random variables.
Prove linearity of expectations of one or two continuous random variables.
Prove the law of iterated expectation for jointly continuous random variables.
TOPIC: Linearity of expectations QUESTION: The random variable X is known to satisfy E[X] = 2 and E[X2] = 7. Find the expected value of 8−X and of (X−3)(X+3). a) E[8−X]= b) E[(X−3)(X+3)]=
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
Prove that any two independent random variables are uncorrelated.
6. (a) Given that X and Y are continuous random variables, prove from first principles that: (b) The random variable X has a gamma distribution with parameters-: 3 and A-2 . Y is a related variable with conditional mean and variance of =x)= Calculate the unconditional mean and standard deviation of Y. (c) Suppose that a random variable X has a standard normal distribution, and the conditional distribution of a Poisson random variable Y, given the value ol XOx, has...
(8) 16 pts] For two random variables X and Y, and for constants a,b,c,d R, prove that Var (aX + b) + (cY + d)] = a2 VarlX) + cWarM + 2acCoolx, y In crafting your argument, you are allowed to use any properties of expectations and/or variances that we covered in lecture.
a. Suppose X and Y are continuous random variables with joint denisty f(x,y). Prove that the density of X+Y is given by: Use part (a) to show that if X,Y are independent and standard Gauss-ian (i.e.N(0,1)) then X+Yi s centered Gaussian with variance 2 that is N(0,2). fx+r(t) = { $(8,6 – u)dt
2. Let X and Y be two random variables with a joint distribution (discrete or continuous). Prove that Cov(X,Y)= E(XY) - E(X)E(Y). (15 points) 3. Explain in detail how we can derive the formula Var(X) = E(X) - * from the formula in Problem 2 above. (Please do not use any other method of proof.) (10 points)
6. (a) State the definition of the covariance Cov(x,Y) of two random variables X and Y. (b) Consider the two continuous random variables X and Y of Ques- tion 2. with joint density f(x, y) otherwise i. Find μχ.y the expectations of X, Y respectively.
(a) Suppose that X, Y and Z are random variables whose joint distribution is continuous with density fxyz. Write down appropriate definitions of of (i) fxyz, density of the joint distribution of X and Y given Z, and (ii) fxyz, density of the distribution of X given both Y and Z. Assuming the expectations exist, prove the tower property: E[E[X|Y, 2]|2] = E[X|2], by expressing both sides using the densities you have defined. Suppose that X and Y are independent...