Question

Let X_{1} , X_{2 ,} X_{3} ,
X_{4} be a random sample of size 4 from a geometric
distribution with p = 1/3.

A) Find the mgf of Y = X_{1} + X_{2} +
X_{3} + X_{4}.

B) How is Y distributed?

Answer #1

**Answer:-**

**Given That:-**

Let be a random sample of size 4 from a geometric distribution with p = 1/3

**a)Find the mgf of
**.?

We know that if are the observation of a random sample from the distribution with the moment - generating function , then the moment - generating function is . Find the moment - generating function of the geometric distribution with p=1/3.

Find the m.g.f of .

**b)How is Y distributed.?**

The m.g.f of Y indicates Y has a negative binomial distribution with r = 4, p = 1/3.

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 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)

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, 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?

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}...

Find the number of solutions to
x1+x2+x3+x4=16 with
integers x1 ,x2, x3, x4
satisfying
(a) xj ≥ 0, j = 1, 2, 3, 4;
(b) x1 ≥ 2, x2 ≥ 3, x3 ≥ −3,
and x4 ≥ 1;
(c) 0 ≤ xj ≤ 6, j = 1, 2, 3, 4

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, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

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

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...

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