Question

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.)

Answer #1

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 ,
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?

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 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 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
distribution with variance σ^2. Let S^2 be the sample variance.
Show that E(S^2)=σ^2.

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ).
Let Yn be the maximum of X1, X2, ..., Xn.
(a) Give the pdf of Yn.
(b) Find the mean of Yn.
(c) One estimator of θ that has been proposed is Yn. You may
note from your answer to part (b) that Yn is a biased estimator of
θ. However, cYn is unbiased for some constant c. Determine c.
(d) Find the variance of cYn,...

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?

Let X1, X2, · · · , Xn be a random sample from the distribution,
f(x; θ) = (θ + 1)x^ −θ−2 , x > 1, θ > 0. Find the maximum
likelihood estimator of θ based on a random sample of size n
above

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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