. Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from an exponential distribution with parameter θ = 1.
(a) Find the pdf of Yr.
(b) Find the pdf of U = e −Yr .
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 < : : : < Y9 be the order statistics of 9 independent draws from an exponential distribution that has a mean of 2. (1) Find the PDF of Y2. (2) Compute P[Y9 < 1].
Let Y1, Y2, ..., Yn denote a random sample from an exponential distribution with mean θ. Find the rejection region for the likelihood ratio test of H0 : θ = 2 versus Ha : θ ≠ 2 with α = 0.09 and n = 14. Rejection region =
Let Y1, Y2, . . . , Yn be independent random variables with Exponential distribution with mean β. Let Y(n) = max(Y1,Y2,...,Yn) and Y(1) = min(Y1,Y2,...,Yn). Find the probability P(Y(1) > y1,Y(n) < yn).
Let Y1, Y2,. . , Yn be a random sample from the population with pdf f(u:)elsewhere (a) If WIn Yi, show that W, follows an exponential distribution with mean 1/0. (b) Show that 2θΣηι W, follows a χ2 distribution with 2n degrees of freedom. (c) It turns out that if X2 distribution with v degrees of freedom, then E( Use this to show
Let Y1< Y2< Y3< Y4< Y5 be the order statistics of n=5 independent observations from the exponential distribution with mean= 1. determine P(Y1>1) and find the pdf of Y5
From 6.3-3. Let Y1 < Y2 < . . . < Y9 be the order
statistics of 9 independent draws from an exponential distribution
that has a mean of 2.
(1) Find the PDF of Y2.
(2) Compute P[Y9 < 1].
From 6.3-3. Let Yǐ < ½ < . . . < y) be the order statistis of 9 independent draws from an exponential distribution that has a mean of 2. (1) Find the PDF of Y2 2) Compute PY,1
Let Y1 , Y2 , . . . , Yn denote a random sample from the uniform
distribution on the interval (θ, θ+1). Let
a. Show that both ? ̂1 and ? ̂2 are unbiased estimators of
θ.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
Let Y1,Y2, …… Yn be a random sample from the distribution f(y) = θxθ-1 where 0 < x < 1 and 0 < θ < ∞. Show that the maximum likelihood estimator (MLE) for θ is