(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If you were taking a random sample of size n (n=2m+1 odd) from Uni(0,1)

How do you find the mean and variance of the sample median?

2. Relevant equations

In order to find the mean and variance of the sample median you need to start with the sample median itself. Using this equation:

ƒ_{xmedian}(x) = ƒ_{x(2m+1)}(x) = [itex]\frac{(2m+1)!}{m!m!}[/itex]*ƒ(x)*[F(x)]^{m}*[1-F(x)]^{m}

Where ƒ(x) is the pdf of the Uni(0,1) ~ Uni(a,b)

ƒ(x) = [itex]\frac{1}{b-a}[/itex] This becomes = 1

Where F(x) is the cdf of the Uni(0,1)

F(x) = [itex]\frac{x-a}{b-a}[/itex] This becomes x

So,

ƒ_{xmedian}(x)

= ƒ_{x(2m+1)}(x)

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*(1)*x^{m}*(1-x)^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*x^{m}*(1-x)^{m}

3. The attempt at a solution

Above is part of the attempt.

Now as for the mean and variance of thesample median

Mean

E(x) = x*ƒ_{xmedian}(x)*dx

= [itex]\int[/itex][itex]^{1}_{0}[/itex] x*[itex]\frac{(2m+1)!}{m!m!}[/itex]*x^{m}*(1-x)^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x*x^{m}*(1-x)^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x^{m+1}*(1-x)^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x*x^{m}*(1-x)^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x*(x^{m}*(1-x)^{m})

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x*[x*(1-x)]^{m}

= [itex]\frac{(2m+1)!}{m!m!}[/itex]*[itex]\int[/itex][itex]^{1}_{0}[/itex] x*[(x-x^{2})]^{m}

-> Continue with integration by parts.

Edit: Above has been corrected

So my question is: Have I done everything above correctly and how would I continue? Is there something I'm missing with this Uniform distribution because that integral doesn't seem to want to simplify.

Variance

Var(x) = E(x^{2}) - [E(x)]^{2}

And similarly for variance will follow.

Thanks!

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# Random sample of size n (n odd) from Uni(0,1)

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