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10. Suppose Y1, ..., Yn are a random sample from a population with density fylu(u) =...
Question 3 [25] , Yn denote a random sample of size n from a Let Y,...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
. 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...
Q- If Y1,…,Yn is an i.i.d. random sample from a population with mean μY and variance...
Q- If Y1,…,Yn is an i.i.d. random sample from a population with mean μY and variance σ2Y, which of the following is not true? 1) Y1,…,Yn are identically distributed random variables 2) Y1,…,Yn are mutually independent random variables 3) Var(Y¯)=σ2Y 4) E(Y¯)=μY
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un-...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
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 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 is an iid sample from a Pareto population distribution described by...
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
. 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...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
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 is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
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