Question

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.

Answer #1

95% confidence interval for μ1 − μ2 ( -6.97, 8.57)

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, Y1), . . . ,(Xn, Yn), be a random sample from a
bivariate normal distribution with parameters µ1, µ2, σ2 1 , σ2 2 ,
ρ. (Note: (X1, Y1), . . . ,(Xn, Yn) are independent). What is the
joint distribution of (X ¯ , Y¯ )?

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, · · · , Xn (n ≥ 30)
be i.i.d observations from N(µ1,
σ12 ) and Y1, Y2, · · ·
, Yn be i.i.d observations from N(µ2,
σ22 ). Also assume that X's and Y's are
independent. Suppose that µ1, µ2,
σ12 ,
σ22 are unknown. Find an
approximate 95% confidence interval for
(µ1µ2).

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

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

Let θ > 1 and let X1, X2, ..., Xn be a random sample from the
distribution with probability density function f(x; θ) = 1/xlnθ , 1
< x < θ.
c) Let Zn = nlnY1. Find the limiting distribution of Zn.
d) Let Wn = nln( θ/Yn ). Find the limiting distribution of
Wn.

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 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

Suppose that
X1,
X2, ,
Xn
and
Y1,
Y2, ,
Yn
are independent random samples from populations with means
μ1
and
μ2
and variances
σ12
and
σ22,
respectively. It can be shown that the random variable
Un =
(X −
Y) − (μ1 −
μ2)
σ12 +
σ22
n
satisfies the conditions of the central limit theorem and thus
that the distribution function of
Un
converges to a standard normal distribution function as
n → ∞.
An experiment is designed to test...

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