Question

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

Answer #1

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.

5. Let a random sample, X1, X2, ..., Xn of size n = 10 from a
distribution that is N(μ1, σ2 ) give ̄x = 4.8 and s 2+ 1 = 8.64 and
a random sample, Y1, Y2, ..., Yn of size n = 10 from a distribution
that is N(μ2, σ2 ) give y ̄ = 5.6 and s 2 2 = 7.88. Find a 95%
confidence interval for μ1 − μ2.

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

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 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, . . . , Xn be a random sample from the normal
distribution N(µ, 36). (a) Show that a uniformly most powerful
critical region for testing H0 : µ = 50 against H1 : µ < 50 is
given by C2 = {x : x ≤ c}. Find the values of c for α = 0.10.

Let X1 and X2 be a sample from a uniform distribution on [0, 1]
and let Y1 = min{X1, X2}, Y2 = max{X1, X2}. Find fY1 (y1|Y2 =
y2).
A. 1/ 2
B. 1 / 2y2
C. 1 /y2
D. 2
E. 1

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?

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.

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

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