(10%) Does there exist uniform distribution X with E[X] Var[X]? Explain! 1. = (10%) Does there...
1. Suppose (x, Y) has bivariate normal distribution, E(x) E(Y)- 0, Var(X) σ , Var(Y) σ and Correl(X, Y) p. Calculate the conditional expectation E(X2|Y).
S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let Yn such that Y,,-, Y almost surely as n-> oo? S2-R be a random variable on a probability space (LF, P) with the uniform distribution on [1-1,T+름 . Does there exist a random variable Y : Ω → R For each n E N, let...
Let x be a continuous random variable. Prove var(x) . (assuming var(x) exist)
1. In cases (a) -(e) below, determine if convergence in distribution takes place for the respective sequences of RVs {Xn) as noo. If it does, find thoe limiting distribution. Explain your answers. (a) X U(0, (cos n)2) (the uniform distributions on the intervals shown) (b) ~ γ(1, ln n) (the garnma distributions; parameters: shape/scale). (c) XnU(v,). d X(normal distributions). (e) Xn (,-P1 (the mixture distribution of the resp. uniform and Poisson distributions, with the weights as shown). n+1 +1 Vn...
Obtain E(Z|X), Var(Z|X) and verify that E(E(Z|X)) =E(Z), Var(E(Z|X))+E(Var(Z|X)) =Var(Z) 3. Let X, Y be independent Exponential (1) random variables. Define 1, if X Y<2 Obtain E (Z|X), Var(ZX) and verify that E(E(Zx)) E(Z), Var(E(Z|X))+E(Var(Z|X)) - Var(Z)
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y. Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Suppose X ∼ N(0, 1). (1) Explain the density of X in terms of the diffusion process. (2) Calculate E(X), E(X^2 ), and Var(X). (3) Let Y = µ + σX. Calculate E(Y ) and Var(Y ). Find the density of Y.
7. Calculate the mgf, E(x), and Var(x) of the following nine distributions: Exponential Distribution
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X. Uniform (a,...