Q has Beta[a,b] (a=6, b=5)
(X|Q=q) i.i.d Geom[q] therefore E[X|Q] = [1-Q / Q] Var[X] = [1-Q / Q^2]
6.1 E[X] = ?
6.2 Var[X] = ?
Q has Beta[a,b] (a=6, b=5) (X|Q=q) i.i.d Geom[q] therefore E[X|Q] = [1-Q / Q] Var[X] =...
EXERCISE 3. Suppose X;'s are i.i.d. Beta B(3,6) (a) Compute E(5Xị - X1X2 – 4); Inn
2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a > 0,6 > 0. The beta distribution has density f(x) = p139–(1 – x)6–1, for 0 < x < 1. If X has the beta distribution, show that E(X) = B(a+1,6)/B(a,b). What is Var(X)?
7. Let x(t) be a Poisson process having rate 6 5. a) P(X(1)=2] b) P(X(2) = 31x(1)s 2] c) P(X(2) = 31x(4) = 5] d) EIX(1)] e) Var[X(1)] (1D 7. Let x(t) be a Poisson process having rate 6 5. a) P(X(1)=2] b) P(X(2) = 31x(1)s 2] c) P(X(2) = 31x(4) = 5] d) EIX(1)] e) Var[X(1)] (1D
E(X) 2 and EX(X - 1))-5. Find Var(X)
1 pts Question 3 Var(x2) 5, and E(x4)= 14. What is var(X) (the answer is an integer)? You are told that E (X) = 1. 1 pts Question 3 Var(x2) 5, and E(x4)= 14. What is var(X) (the answer is an integer)? You are told that E (X) = 1.
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some distribution with mean u and finite variance. Define the sample variance S2 = (n-1)-1 _, (Xi - X)2. We will show that S2 is an unbiased estimator of the population variance Var(X1). (i) Show that ) = 0. (ii) Show that [ŠX – 1908–) -0. ElCX –po*=E-* (Šx--) == "Varex). x:== X-X+08 – ) Lx - X +2Zx - XXX - 1) + X...
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)
For the random variables X and Y having E(X) = 1, E(Y) = 2, Var (X) = 6, Var (Y) = 9, and Pxy = -2/3. Find a) The covariance of X and Y. b) The correlation of X and Y. c) E(X2) and E(Y2).
(1 point) Para una variable aleatoria X, supongamos que E[X] = 5 y Var(X) = 10. entonces (a) E[( 5 x)?] = (b) Var(2 + 6X) =
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful