Question

Suppose that X_{i} are IID normal random variables with
mean 5 and variance 10, for i = 1, 2, ... , n.

(a) Calculate P(X_{1} < 6.2), i.e., the probability that
the first value collected is less than 6.2.

(b) Suppose we collect a sample of size 2, X_{1} and
X_{2}. What is the probability that their sample mean is
greater than 6.3?

(c) Again, suppose we collect two samples (n=2), X_{1} and
X_{2}. What is the probability that their sum is greater
than 11.4? (Hint: P(X_{1} + X_{2} > 11.4) = P( X
> 5.7).

(d) Suppose we collect a sample size of n=100. What is the
probability that the sample mean is between 4.53 and 5.32?

Answer #1

Let X1, X2 be two normal random variables each with population
mean µ and population variance σ2. Let σ12 denote the covariance
between X1 and X2 and let ¯ X denote the sample mean of X1 and X2.
(a) List the condition that needs to be satisﬁed in order for ¯ X
to be an unbiased estimate of µ. (b) [3] As carefully as you can,
without skipping steps, show that both X1 and ¯ X are unbiased
estimators of...

Suppose that X1,..., Xn∼iid Geometric(p).
(a) Suppose that p has a uniform prior distribution on the
interval [0,1]. What is the posterior distribution of p?
For part (b), assume that we obtained a random sample of size 4
with ∑ni=1 xi = 4.
(b) What is the posterior mean? Median?

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?

X1,X2, . . . is a sequence of iid
Bernoulli (1/2) random variables. Consider
the random sequence Y_n = X_1 +· · ·+X_n.
(a) What is limn→∞ P[|Y_2n − n| ≤ (n/2)^1/2?
(b) What does the weak law of large numbers
say about Y2n?
Could I get a little clarification with this problem?

Let X1, X2,... be a sequence of
independent random variables distributed exponentially with mean 1.
Suppose that N is a random variable, independent of the Xi-s, that
has a Poisson distribution with mean λ > 0. What is the expected
value of X1 + X2 +···+
XN2?
(A) N2
(B) λ + λ2
(C) λ2
(D) 1/λ2

Question 7) Suppose X is a Normal random variable with with
expected value 31 and standard deviation 3.11. We take a random
sample of size n from the distribution of X. Let X be the sample
mean. Use R to determine the following:
a) Find the probability P(X>32.1):
b) Find the probability P(X >32.1) when n = 4:
c) Find the probability P(X >32.1) when n = 25:
d) What is the probability P(31.8 <X <32.5) when n =
25?...

Let X1, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

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...

Suppose X is a Normal random variable with with expected value
31 and standard deviation 3.11. We take a random sample of size n
from the distribution of X. Let X be the sample mean. Use R
to determine the following:
a) What is the standard deviation of X when n =
19?
b) What is the probability that X1 + X2 +
... +X20 > 630?
PLEASE ANSWER IN R SCRIPT

For
Questions 6 - 8, let the random variable X follow a Normal
distribution with variance σ2 = 625.
Q6. A random sample of n = 50 is obtained with a sample mean, X-Bar
of 180.
What is
the probability that population mean μ is greater than 190?
a.
What is Z-Score for μ greater than 190 ==>
b.
P[Z > Z-Score] ==>
Q7. What
is the probability that μ is between 198 and 211?
a. What
is Z-Score1 for...

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