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Help me find the probability density function

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mhamini

mhamini

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Suppose X1, . . . ,Xn are independently and identically from the uniform distribution on [0, 1]. Find the probability density function of Y = min[X1, X2, ... , Xn].
I do not know how to formulate this problem. I know that the pdf has to be some integral, but no clue so far.
 

Re: Finding pdf

Here it is:

Suppose that f(x) and F(x) are the density function and the cumulative distribution function, respectively. Then,

f(x)=1, when 0<x<1, and 0 otherwise.

You can easily get F(x).
Now, for any y,

P(Y>y) = P(min[X₁,X₂,X₃,...X_{n}]>y)
= P(X₁>y,X₂>y,X₃>y,...X_{n}>y)
= P(X₁>y)P(X₂>y)...P(X_{n}>y)
= (1-F(y))(1-F(y))...(1-F(y))
= (1-F(y))ⁿ.

Therefore,

P(Y<y) = 1-P(Y>y)
= 1-(1-F(y))ⁿ.

Your density function is

n(1-F(y))ⁿ⁻¹.
 

    mhamini

    mhamini

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Re: Finding pdf

this is a problem called order statistics, which is used in another sense in communication systems ( to find the maximum of a bunch of signals, then the next ma, and so on)
in my idea i think it's best to read aboutit in the book by papoulis and unikrishna " probability and random variables" chapter 8
 

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