n-th percentile

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June 7th 2010, 09:58 AM
losm1

n-th percentile

I need to find 10th percentile of random variable X with PDF defined with:

$<br /> f(x) = \left\{\begin{array}{cc}c(x-x^3)&0< x < 1\\0&x\le 0 \vee x\ge 1 \end{array}\right.<br />$

where $c\in \mathbb{R}$ is const that should be found too.
By integrating f from 0 to $x$ I got cumulative dist. function $F(x) = c \frac{2x^2 - x^4}{4}$ , so $c=4$ .

First, is this approach OK, and second, how can I calculate required percentile?

Thanks guys
June 7th 2010, 10:36 AM
SpringFan25

Your method for finding the CDF is correct(but i didn't check you did the integral properly).

the 10 percentile $x_{10}$ satisfies:

$F(x_{10}) = 0.1$
June 7th 2010, 10:46 AM
losm1

I'm little bit confused with notation $x_{10}$ , can you write complete calculation pls?
June 7th 2010, 11:03 AM
SpringFan25

i have just called the 10th percentile $x_{10}$

The definition of a percentile is:

$P(X<x_{10}) = 0.1$
$F(x_{10}) = 0.1$

So find the value of x that satisfies

$2x^2 - x^4 = 0.1$

Can you finish from there?

Hint: this is a quadratic in $x^2$
June 7th 2010, 11:16 AM
losm1

Ok, thanks. So we take a root that is in interval $0<x<1$ (we have 4 solutions, one in that interval)?
June 7th 2010, 11:28 AM
SpringFan25

yes, since x can not be outside that interval

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