Let X; be n IID U(0, 1) random variables. What are the mean and
variance of the minimum-order and maximum-order statistics?
PLEASE SHOW ALL WORK AND FORMULAS USED
Let X1,…, Xn be a sample of iid N(0, ?) random
variables with Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) the asymptotic variance of the MLE of ?.
d) the MLE of SD(Xi ) = √ ?.
Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR in Y
= X(n)? (b) Find the UMP size-α test for H0 : θ ≤ θ0 vs H1 : θ >
θ0. (c) Find the UMP size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0.
(d) Letting R1 be the rejection region for the test in part (b) and
R2 be the rejection region for the test in part...
Let ?1, ?2,…. . , ?? (n random variables iid) as a
variable whose pdf is continuous and uniform over the interval [? -
1; ? + 3].
(1) Determine the estimator of the moments method.
(2) Is this estimator unbiased? What is its variance?
(3) Find the maximum likelihood estimator (VME) for this
setting. Is it unique?
Let (Un, U, n>1) be asequence of random variables such that
Un and U are independent, Un is N(0, 1+1/n), and U is N(0,1), for
each n≥1.
Calculate p(n)=P(|Un-U|<e), for all e>0.
Please give details as much as possible
Let U and V be vector spaces, and let L(V,U) be the set of all
linear transformations from V to U. Let T_1 and T_2 be in
L(V,U),v be in V, and x a real number. Define
vector addition in L(V,U) by
(T_1+T_2)(v)=T_1(v)+T_2(v)
, and define scalar multiplication of linear maps as
(xT)(v)=xT(v). Show that under
these operations, L(V,U) is a vector space.
1. Let X and Y be independent U[0, 1] random variables, so that
the point (X, Y) is uniformly distributed in the unit square.
Let T = X + Y.
(a) Find P( 2Y < X ).
(b). Find the CDF F(t) of T (for all real numbers t).
HINT: For any number t, F(t) = P ( X <= t) is just the area
of a part of the unit square.
(c). Find the density f(t).
REMARK: For a...
Let X1, X2, . . . be iid random variables following a uniform
distribution on the interval [0, θ]. Show that max(X1, . . . , Xn)
→ θ in probability as n → ∞
Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a)
Let U ∈ Rn×n be an orthogonal matrix (UTU = UUT = In) and nd the
distribution of UTx. Let y = (y1,...,yn) ∼ N(0,Σ) be a MVN random
vector in Rn. Let Σ = UΛUT be the spectral decomposition of
Σ.
(b) Someone claims that the diagonal elements of Λ are nonnegative.
Is that true?
(c) Let z = UTy and nd the distribution of...