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4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show...
Please answer all the parts neatly with all details.
4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
1. Let X1, ..., Xn, Y1, ..., Yn be mutually independent random variables, and Z = + Li-i XiYi. Suppose for each i E {1,...,n}, X; ~ Bernoulli(p), Y; ~ Binomial(n,p). What is Var[Z]?
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
(d) Show that if and are distinct eigenvalues of a square matrix A, x = (x1; x2; : : : ; xn) 2 E[A], y = (y1; y2; : : : ; yn) 2 E[A] then: x; y = x1y1 + x2y2 + + xnyn = 0:
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
Suppose that X1, X2,.... Xn and Y1, Y2,.... Yn are independent random samples from populations with the same mean μ and variances σ., and σ2, respectively. That is, x, ~N(μ, σ ) y, ~ N(μ, σ ) 2X + 3Y Show that is a consistent estimator of μ.