# Independent Uniform Random Variables

• Nov 23rd 2009, 10:01 AM
sirellwood
Independent Uniform Random Variables
If X1,X2, . . . ,Xn are independent U(0, 1) random variables and Y=min (X1,X2, . . . ,Xn) has PDF given by

fY(x) = { $n(1-x)^{n-1}$, 0<=x<=1}, 0, otherwise

Identify this distribution, and hence state E[Y] and Var(Y).

Im thinkin it looks similar to a binomial distribution? Am i on the right lines?
• Nov 23rd 2009, 06:22 PM
mr fantastic
Quote:

Originally Posted by sirellwood
If X1,X2, . . . ,Xn are independent U(0, 1) random variables and Y=min (X1,X2, . . . ,Xn) has PDF given by

fY(x) = { $n(1-x)^n-1)$, 0<=x<=1}, 0, otherwise Mr F says: What you have posted here is wrong. It's not even a pdf ....

Identify this distribution, and hence state E[Y] and Var(Y).

Im thinkin it looks similar to a binomial distribution? Am i on the right lines?

There is a (simple) mistake in your pdf which I'll leave you to figure out and fix.

The correct pdf has a beta distribution (and you will need to determine what value each parameters has).
• Nov 24th 2009, 04:42 AM
sirellwood
Ah thank you mr fantastic! its edited now!

So i first need to find the sample mean and variance right? Because it is uniformally distributed, is the sample mean just (x1+xn)/2?

Im getting a little confused with the word "min"(x1,x2....) in the Y term.
• Nov 24th 2009, 10:57 PM
matheagle
minimum of X1,...,Xn is the smallest of these rvs
It's also known as the smallest order stat.
It couldn't be a binomial since the underlying distribution here is continuous (uniform)

$F_Y(a)=P(Y\le a)=1-P(Y>a)$ for any 0<a<1

$=1- P(X_1>a)P(X_2>a)\cdots P(X_n>a)$

$=1- (1-a)^n$

So $f_Y(a)={d\over da}F_Y(a)=n(1-a)^{n-1}$

THIS is a Beta with parameters 1 and n.
Look that up for your mean and variance.
• Nov 25th 2009, 04:03 AM
mr fantastic
Quote:

Originally Posted by sirellwood
Ah thank you mr fantastic! its edited now!

So i first need to find the sample mean and variance right? Because it is uniformally distributed, is the sample mean just (x1+xn)/2?

Im getting a little confused with the word "min"(x1,x2....) in the Y term.

The min business is irrelevant (unless you want to derive the given pdf). The bottom line is that you're given a pdf and asked to identify it and hence get it's mean and variance. Note that you can check your answers by calculating the mean and variance directly from the pdf.