Question

- Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2.

Consider the following estimator of μ: 1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased estimator for the mean? What is the variance of the estimator? Can you find a more efficient estimator?) ( 10 Marks)

Answer #1

Let X1 , X2 , X3 ,
X4 be a random sample of size 4 from a geometric
distribution with p = 1/3.
A) Find the mgf of Y = X1 + X2 +
X3 + X4.
B) How is Y distributed?

Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...

Let X1, X2, X3 be a random sample of size 3 from a distribution
that
is Normal with mean 9 and variance 4.
(a) Determine the probability that the maximum of X1; X2; X3
exceeds 12.
(b) Determine the probability that the median of X1; X2; X3 less
than
10.
(c) Determine the probability that the sample mean of X1; X2;
X3
less than 10. (Use R or other software to find the
probability.)

Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let
S = X1 + X2 + X3 + X4. Suppose we know that S is a Chi-Square
random variable with 2 degrees of freedom. What
is the distribution of each of the Xi?

5. Consider a simple case with only four independently and
identically distributed (iid) observations, X1, X2, X3, X4, on a
random variable X. Consider these two estimators:
µˆ1 = 1/12 (2X1 + 4X2 + 4X3 + 2X4), µˆ2 = 1/12 (X1 + 5X2 + 5X3 +
X4).
a Show that each is unbiased, and that one is more efficient
than the other.
b Show that the usual sample mean is more efficient than either.
Explain why the others given above...

Let X1, X2, X3 be a random sample of size 3 from a distribution
that is Normal with mean 9 and variance 4.
(a) Determine the probability that the maximum of X1; X2; X3
exceeds 12.
(b) Determine the probability that the median of X1; X2; X3 less
than 10.
Please I need a solution that uses the pdf/CDF of the
corresponding order statistics.

A random sample of 100 observations Denote byX1,X2,...,X100,
were selected from a population with a mean μ= 40 and variance σ2=
400.
Determine P(38< ̄X <43), where ̄X=
(X1+X2+...+X100)/100.

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

#1 A sample of 4 observations (X1 = 0.4,
X2 = 0.6, X3 = 0.7, X4 = 0.9) is
collected from a continuous distribution with pdf
(a) Find the point estimate of θ by the Method of
Moments.
(b) Find the point estimate of θ by the Method of
Maximum Likelihood. Use two decimal places.

Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?

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