(7) 15 ptsl Let Y - a +bX +U, where X and U d b are are randon variables and a an constants. Assume that E[U|X] 0 and Var u|X] - X2. (a) Is Y a random variable? Why? (b) Is U independent of X? Why? (c) Show that Eu0 and Var[uEX2] (d) Show that E[Y|X- a bX, and that E[Y abEX]. (e) Show that VarlyX] = X2, and that Varly-p?Var(X) + EX2].
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1). Obtain fy and compute E[Y] (b) Now instead assume that Y ~ U(0,1/2) and that X is a continuous random variable. Explain carefully why it is possible to choose fx such that fx (2) = 0 whenever 21 > 1. Obtain an expression linking fx(2) to fx(-x) for 3 € (-1,1). Show that E[X] = -2/3 + 2 S xfx(x) dx. Using your expression...
Question 34 In the exercise below, let U = {x|XE N and x < 10} A = {x|x is an odd natural number and x < 10} B = {x|x is an even natural number and x < 10} C = {x|x € N and 3 <x<5} Find the set. Во С {4} {1, 2, 3, 4, 5, 6, 7, 8, 9} {2, 4, 6, 8, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
Let $(x) = 2x2 and let Y = $(X). assume that Y ~ U(0,1/2) and that X is a continuous random variable. fx(x) = 0 whenever |2| > 1. Obtain an expression linking fx(x) to fx(-x) for xe (-1,1). Show that E[X] = -2/3 + 28. xfx(x) dx. Using your expression linking fx(x) and fx(-x), obtain an upper bound for E[X] and a pdf fx for which this bound is attained. [10]
7.4 Let X ~ U(-1,1) and Y = x2. a. What are the density, the distribution function, the mean, and the variance of Y: b. What is Pr[Y < 0.5]? 7.5 Let X – U(0,1), and let Y = eax for some a > 0. What are the density, the distribution function, the mean, and the variance of Y?
- Let X, 1.2.4. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X]. Sketch the pdf fxı. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it...
3. Let X.. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X,. Sketch the pdf fx. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it can...
Let X1 and X2 be independent random variables so X1~ N(u,1) and X2 N(u,4) Where u R a) Show that the likelihood for , given that X1 = x1 and X2 = xz is 8 4T b) Show, that the maxium likelihood estimate for u is 4x1+ x2 и (х, х2) e) Show that СтN -("x"x) .я d) and enter a formula for the 95% confidence interval for Let X1 and X2 be independent random variables so X1~ N(u,1) and...
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X = = | 12 (a) Find the distribution of Y1 = X1 + X2. (b) Let Y2 = X1 +aX2. Find value a such that Yį and Y2 are independent.
3. (7 points) Let u(x, y) be the steady-state temperature u(r, y) in a rectangular plate whose vertical r0 and 2 are insulated. When no heat escapes, we have to solve the following the boundary value problem: a(z,0) = 0, u(z,2) = x, 0 < x < 2 (a) By setting u(x, g) -X(x)Y(u), separate the equation into two ODE 0 What ane the sewr homdany condiome hoald Xe) watiy (37)2. (c) Find x(r) for the case when λ-0 and...