Random samples Y_1,Y_2,Y_3,\dots,Y_nY1,Y2,Y3,…,Yn were selected
from an X distribution with mean µ and variance σ2.
a) If Z_1=(Y_1-μ)/σZ1=(Y1−μ)/σ, state the distribution for Z1 . (10
marks)
Let
i. Can we conclude that the distribution answer in 1a) is same with the distribution for Z_1^2Z12 Justify your answer. (10 marks)
ii. If Z_1^2+Z_2^2+Z_3^2+⋯+Z_n^2Z12+Z22+Z32+⋯+Zn2 = W, can we assume W has a similar distribution as X. Explain your answer. (10 marks)
Random samples Y_1,Y_2,Y_3,\dots,Y_nY1,Y2,Y3,…,Yn were selected from an X distribution with mean µ and variance σ2. a)...
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
Let Y1, Y2, , Yn be independent, normal random variables, each
with mean μ and variance σ^2.
(a) Find the density function of
f Y(u) =
(b) If σ^2 = 25 and n = 9, what is the
probability that the sample mean, Y, takes on a value that is
within one unit of the population mean, μ?
That is, find P(|Y − μ| ≤ 1). (Round your answer to four decimal
places.)
P(|Y − μ| ≤ 1) =
(c)...
2. (10pts) The following 16 random samples; 5.33, 4.25, 3.15, 3.70, 1.61, 6.40, 3.12, 6.59,3.53, 4.74, 0.11, 1.60, 5.49, 1.72, 4.15, 2.30, came from normal distribution with mean μ and variance σ2, i.e., Xi, X2' .. . , X16 ~ N(μ, σ*), with the density function (a) (4pts) Find the maximum likelihood estimates of μ and σ2, denoted with μ and σ2. (b) (4pts) Based on above μ and σ2, construct 95% confidence intervals for μ and σ2 separately. (c)...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...