Let Xi's be independent and identically distributed
Poisson random variables for 1 ≤ i ≤ n....
Let Xi's be independent and identically distributed
Poisson random variables for 1 ≤ i ≤ n. Derive the distribution for
the summation of Xi from 1 to n. (Without using MGF)
Let ?1 , ?2 , ... , ?? be independent, identically distributed
random variables with p.d.f. ?(?) = ???−1, 0 ≤ ? ≤ 1 . c) Show that
the maximum likelihood estimator for ? is biased, and find a
function of the mle that is unbiased. (Hint: Show that the random
variable −ln (??) is exponential, the sum of exponentials is Gamma,
and the mean of 1/X for a gamma with parameters ? and ? is 1⁄(?(? −
1)).) d)...
Let ?1,?2,…,??be independent, identically distributed random
variables with p.d.f. ?(?) = ??^?−1,0 ≤ ? ≤ 1 .
c. Show that the maximum likelihood estimator for ? is biased,
and find a function of the mle that is unbiased. (Hint: Show that
the random variable −ln (??) is exponential, the sum of
exponentials is Gamma, and the mean of 1/X for a gamma with
parameters ? and ? is 1 / (?(? − 1)).
d. Is the estimator you found in...
Let X1, X2, . . . be a sequence of independent and identically
distributed random variables where the distribution is given by the
so-called zero-truncated Poisson distribution with probability mass
function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2,
3...
Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is
independent of the Xi ’s.
1) Show that Y = X1
+X2 + ... + XN has a Poisson distribution
with mean nλ.
Suppose Y1,Y2, .. ,Y8 are independent and identically
distributed as Poisson random variables with mean
lambda.
a) Derive the most powerful test for testing Ho: lambda = 2, Ha:
lambda = 3. Carefully show all work involved in the derivation.
i) Give the form of the test. (In other words, for what general
values of Y1,Y2, .. ,Y8 will Ho be rejected?)
ii) Describe the rejection region as carefully as possible if
alpha <= .05 (and is as close as...
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). 5
MARKS
(ii) Now suppose that X and Y are independent and identically
distributed N(1, 2.56) random variables. What is P(|X + Y − 2| ≥ 1)
exactly? Briefly, state your reasoning. 2 MARKS
(iii) Why is the upper bound you obtained in Part (i) so
different...
Let Y1, Y2, Y3, and
Y4be independent, identically distributed random
variables from a population with a mean μ and a variance
σ2. Consider a different estimator of μ:
W
= Y1+ Y2+
Y3+ Y4.
Let Y1, Y2, Y3, and
Y4be independent, identically distributed random
variables from a population with a mean μ and a variance
σ2. Consider a different estimator of μ:
W = 1/8 Y1+ 1/3
Y2+ 1/6 Y3+ 3/8 Y4.
This is an example of a weighted
average of the Yi.
Show...
Let X1, X2,...,
Xnbe independent and identically distributed
exponential random variables with parameter λ .
a) Compute P{max(X1,
X2,..., Xn) ≤ x}
and find the pdf of Y = max(X1,
X2,..., Xn).
b) Compute P{min(X1,
X2,..., Xn) ≤ x}
and find the pdf of Z = min(X1,
X2,..., Xn).
c) Compute E(Y) and E(Z).
Let X1 and X2 be independent identically
distributed random variables with pmf p(0) = 1/4, p(1) = 1/2, p(2)
= 1/4
(a) What is the probability mass function (pmf) of X1
+ X2?
(b) What is the probability mass function (pmf) of
X(2) = max{X1, X2}?
(c) What is the MGF of X1?
(d) What is the MGF of X1 + X2
Let X1 and X2 be independent identically distributed random
variables with pmf p(0) = 1/4, p(1) = 1/2, p(2) = 1/4
(a) What is the probability mass function (pmf) of X1 + X2?
(b) What is the probability mass function (pmf) of X(2) =
max{X1, X2}?
(c) What is the MGF of X1?
(d) What is the MGF of X1 + X2? (Note: The formulas we did were
for the continuous case, so they don’t directly apply here, but you...
A: Suppose two random variables X and Y are independent and
identically distributed as standard normal. Specify the joint
probability density function f(x, y) of X and Y.
Next, suppose two random variables X and Y are independent and
identically distributed as Bernoulli with parameter 1 2 . Specify
the joint probability mass function f(x, y) of X and Y.
B: Consider a time series realization X = [10, 15, 23, 20, 19]
with a length of five-periods. Compute the...