f(y)= 3y^2/theta^3 from 0<y<theta, o otherwise. a) Find the pdf of Y(n)= max(Y1,Y2,...,Yn) b) if n=11...
f(y)= 3y^2/theta^3 from 0<y<theta, o otherwise.
a) Find the pdf of Y(n)= max(Y1,Y2,...,Yn)
b) if n=11 find E(Y(n))
c) if n=11 find the pdf of the median
Homework Answers
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f(y)= 3y^2/theta^3 from 0<y<theta, o otherwise.
a) Find the pdf of Y(n)= max(Y1,Y2,...,Yn)
b) if n=11...
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution...
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)
7. Let Y1, ...,Yn be a random sample from the population with pdf f(316) = he=1/0,...
7. Let Y1, ...,Yn be a random sample from the population with pdf f(316) = he=1/0, y>0 (a) Find the MOM estimator for 0. (b) Find the MLE of 0. (c) Find the MLE of P(Y < 2). (d) Find the MLE of the median of the distribution.
Let Y1, Y2,. . , Yn be a random sample from the population with pdf f(u:)elsewhere...
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
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution...
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution on [0,k]. a. Determine the density of Y(n) = max(Y1, Y2, …, Yn). b. Compute the bias of the estimator k = Y(n) for estimating k.
. Let Y1 < Y2 < · · · < Yn be the order statistics of...
. 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 .
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0...
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0 otherwise Let U1 = min{Y1, ... ,Yn} and U2 = $1=1 Yį. (a) Show that the pdf of Uı is f(y) = ne-n(y=0)1(y > 0) (b) Show that U1 - 1/n and U2/n - 1 are both unbiased estimators of 0. (c) Find the variance of each of the unbaised estimators in part (b). (d) One of U1,U2 is a sufficient statistic. Which one?...
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3. (20 points) If Y., Y2, 0 Syse, a. find cÝ, where c is a constant, that is an unbiased estimator of θ; and b. show that the varian...
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3. (20 points) If Y., Y2, 0 Syse, a. find cÝ, where c is a constant, that is an unbiased estimator of θ; and b. show that the variance of is less than the Cramér-Rao lower bound for fr (y; 0) c. Why isn't this a violation of the Cramér-Rao inequality?
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3....
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a...
. 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)
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
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)
7. Let Y1, ...,Yn be a random sample from the population with pdf f(316) = he=1/0, y>0 (a) Find the MOM estimator for 0. (b) Find the MLE of 0. (c) Find the MLE of P(Y < 2). (d) Find the MLE of the median of the distribution.
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
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0 otherwise Let U1 = min{Y1, ... ,Yn} and U2 = $1=1 Yį. (a) Show that the pdf of Uı is f(y) = ne-n(y=0)1(y > 0) (b) Show that U1 - 1/n and U2/n - 1 are both unbiased estimators of 0. (c) Find the variance of each of the unbaised estimators in part (b). (d) One of U1,U2 is a sufficient statistic. Which one?...
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3. (20 points) If Y., Y2, 0 Syse, a. find cÝ, where c is a constant, that is an unbiased estimator of θ; and b. show that the variance of is less than the Cramér-Rao lower bound for fr (y; 0) c. Why isn't this a violation of the Cramér-Rao inequality?
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3....
. 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...
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