Question

Suppose X and Y are independent variables with E(X) = E(Y ) = θ, Var(X) = 2 and Var(Y ) = 4. The two estimators for θ, W1 = 1/2 X + 1/2 Y and W2 = 3/4 X + 1/4 Y .

(1) Are W1 and W2 unbiased? (2) Which estimator is more efficient (smaller variance)?

Answer #1

Suppose that X and Y
are random samples of observations from a population with mean μ
and variance σ2.
Consider the following
two unbiased point estimators of μ.
A = (2/3)X + (1/3)Y B
= (5/4)X - (1/4)Y
Find variance of A.
Var(A) and Var(B)
Efficient and unbiased
point estimator for μ is = ?

VERY URGENT !!!! Suppose that X and Y are random samples of
observations from a population with mean μ and variance
σ2.
Consider the following two unbiased point estimators of μ.
A = (7/4)X - (3/4)Y B = (1/3)X +
(2/3)Y
[Give your answers as ratio (eg: as number1 /
number2 ) and DO NOT make any
cancellation]
1. Find variance of A. Var(A)
=(Answer)*σ2
2. Find variance of B. Var(B)
=(Answer)*σ2
3. Efficient and unbiased point...

Suppose that E[X]= E[Y] = mu, where mu is a fixed unknown
number. We have independent simple random samples of size n each
from the distribution of X and Y, respectively. Suppose that Var[X]
= 2*Var[Y]. Consider the following estimators of mu:
m1 = bar{X}
m2 = bar{Y}/2
m3 = 3*bar{X}/4 + 2*bar{Y}/8
where bar{X} and bar{Y} are the sample mean of X and Y values,
respectively. Which of the estimators are unbiased?

1) Suppose that X and Y are two random variables, which may be
dependent and Var(X)=Var(Y). Assume that 0<Var(X+Y)<∞ and
0<Var(X-Y)<∞. Which of the following statements are NOT true?
(There may be more than one correct answer)
a. E(XY) = E(X)E(Y)
b. E(X/Y) = E(X)/E(Y)
c. (X+Y) and (X-Y) are correlated
d. (X+Y) and (X-Y) are not correlated.
2) S.D(X ± Y) is equal to, where S.D means standard
deviation
a. S.D(X) ± S.D(Y)
b. Var(X) ± Var(Y)
c. Square...

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

If X, Y are random variables with E(X) = 2, Var(X) = 3, E(Y) =
1, Var(Y) =2, ρX,Y = −0.5
(a) For Z = 3X − 1 find µZ, σZ.
(b) For T = 2X + Y find µT , σT
(c) U = X^3 find approximate values of µU , σU

Consider two random variables X and Y such that E(X)=E(Y)=120,
Var(X)=14, Var(Y)=11, Cov(X,Y)=0.
Compute an upper bound to
P(|X−Y|>16)

Let X ~ N(1,3) and Y~ N(5,7) be two independent random
variables. Find...
Var(X + Y + 32)
Var(X -Y)
Var(2X - 4Y)

Let X and Y be independent and normally distributed random
variables with waiting values E (X) = 3, E (Y) = 4 and variances V
(X) = 2 and V (Y) = 3.
a) Determine the expected value and variance for 2X-Y
Waiting value µ = Variance σ2 = σ 2 =
b) Determine the expected value and variance for ln (1 + X
2)
c) Determine the expected value and variance for X / Y

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

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