Find expected value, variance and check if estimator is unbiased for ?1, …, ?? ~ (iid)?(2?, 7?)

[Radaguna]

Could you please help me to solve this? If expected value in uniform distribution is equal (a+b)/2, do I just have to do (2? + 7?)/2 = 4.5? and if Var = ((b-a)^2)12, then it’s equal 2.08? Or I’m missing something?

Let ?1, …, ?? ~ (iid)?(2?, 7?)

a) Find ???:?

b) Find ??? ??:?

c) Find the value of ?? for which estimator ?=????:? is an unbiased estimator of the parameter ?

 

[BigBeachBanana]

The question is unclear.
Are you saying [imath]X_i\sim U(2\theta,7\theta)[/imath]
or for [imath]Y=X_1+X_2+\dots+X_N[/imath], [imath]; Y\sim U(2\theta,7\theta)?[/imath]
Please attach a picture of the problem instead.

 

[Radaguna]

The question is unclear.
Are you saying [imath]X_i\sim U(2\theta,7\theta)[/imath]
or for [imath]Y=X_1+X_2+\dots+X_N[/imath], [imath]; Y\sim U(2\theta,7\theta)?[/imath]
Please attach a picture of the problem instead.

Thank you for your answer, here is the picture

 

[BigBeachBanana]

Thank you for your answer, here is the picture
View attachment 32049

I'm not familiar with the notation [imath]X_{n:n}[/imath], is that the sum of the random variables?
If so, then you're looking for
a) [imath]E(Y)=E(X_1+X_2+\dots+X_n)\stackrel{i.i.d}{=}nE(X_i)[/imath], where [imath]X_i\sim U(2\theta,7\theta)[/imath]
Use similar idea for b).

 

[Radaguna]

I'm not familiar with the notation [imath]X_{n:n}[/imath], is that the sum of the random variables?
If so, then you're looking for
a) [imath]E(Y)=E(X_1+X_2+\dots+X_n)\stackrel{i.i.d}{=}nE(X_i)[/imath], where [imath]X_i\sim U(2\theta,7\theta)[/imath]
Use similar idea for b).

Xn:n is the last order statistics, like X1:n is the first order. I don't quite understand what to do with this new variable Y?

 

[BigBeachBanana]

Xn:n is the last order statistics, like X1:n is the first order. I don't quite understand what to do with this new variable Y?

If [imath]X_{n:n}[/imath] is the last order statistics, then are you looking for the expected value of max[imath](X_1,X_2,\dots,X_n)?[/imath]
I'm just unclear what the question is asking for.

 

[Radaguna]

If [imath]X_{n:n}[/imath] is the last order statistics, then are you looking for the expected value of max[imath](X_1,X_2,\dots,X_n)?[/imath]
I'm just unclear what the question is asking for.

Sorry for being not clear enough, but yes, i need to find expected value of max[imath](X_1,X_2,\dots,X_n)[/imath]

 

[BigBeachBanana]

Sorry for being not clear enough, but yes, i need to find expected value of max[imath](X_1,X_2,\dots,X_n)[/imath]

Let [imath]Y=max(X_1,X_2,\dots,X_n)[/imath]. Before you can find the expected value and variance, find the density function first.
\(\displaystyle \Pr(Y\le y)=\Pr(max(X_1,X_2,\dots,X_n)\le y)= \Pr(X_1 \le y\, \cap X_2 \le y\, \cap \dots X_n \le y) \stackrel{i.i.d}{=}\Pr(X_1 \le y)\Pr(X_2 \le y)\dots\Pr(X_n \le y)=y^n\); where [imath]y \in (2\theta,7\theta)[/imath]
What's the CDF of Y? Differentiate to get the PDF of Y.