Question

Suppose that X1 and X2 denote a random sample of size 2 from a gamma distribution, Xi ~ GAM(2, 1/2).

Find the pdf of W = (X1/X2). Use the moment generating function technique.

Answer #1

If X1 and X2 denote random sample of size 2 from Poisson
distribution, Xi is distributed as Poisson(lambda), find pdf of
Y=X1+X2. Derive the moment generating function (MGF) of Y as the
product of the MGFs of the Xs.

Let X1, X2 be a sample of size 2 from the Gamma (Alpha=2, Lamba
= 1/theta) distribution
X1 = Gamma = x/(theta^2) e^(-x/theta)
Derive the joint pdf of Y1=X1 and Y2 = X1+X2
Derive the conditional pdf of Y1 given Y2=y2. Can you name that
conditional distribution? It might not have name

Let X1, X2, . . . , X12 denote a random sample of size 12 from
Poisson distribution with mean θ.
a) Use Neyman-Pearson Lemma to show that the critical region
defined by
(12∑i=1) Xi, ≤2
is a best critical region for testing H0 :θ=1/2 against H1
:θ=1/3.
b.) If K(θ) is the power function of this test, find K(1/2) and
K(1/3). What is the significance level, the probability of the 1st
type error, the probability of the 2nd type...

Let X1,X2,...,X50 denote a random sample of size 50 from the
distribution whose probability density function is given by f(x)
=(5e−5x, if x ≥ 0 0, otherwise If Y = X1 + X2 + ... + X50, then
approximate the P(Y ≥ 12.5).

Let X1, X2 be a random sample of size 2 from the standard normal
distribution N (0, 1). find the distribution of {min(X1, X2)}^2

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.

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 · · · , Xn be a random sample from the distribution
with PDF, f(x) = (θ + 1)x^θ , 0 < x < 1, θ > −1.
Find an estimator for θ using the maximum likelihood

Let X1, X2, . . . Xn be iid
random variables from a gamma distribution with unknown α and
unknown β. Find the method of moments estimators for α and β

Let X1,X2, . . . ,Xn be a random sample of size n
from a geometric distribution for which p is the probability
of success.
(a) Find the maximum likelihood estimator of p (don't use method of
moment).
(b) Explain intuitively why your estimate makes good
sense.
(c) Use the following data to give a point estimate of p:
3 34 7 4 19 2 1 19 43 2
22 4 19 11 7 1 2 21 15 16

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