Wikipedia:Reference desk/Archives/Mathematics/2017 August 27

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August 27

Wikipedia article about adding subtracting dividing and multiplying two PDF

A long long time ago, I asked a question about how to form a new pdf (probability density function) which is a product of two other pdf

PDF_A = PDF_H * PDF_W

Where PDF_A is the Probability Density Function of Area

PDF_H is the Probability Density Function of Height

PDF_W is the Probability Density Function of Width

And the answer given is

Is there a wiki article about how to do this for

PDF_B = Sqrt(PDF_A)

PDF_B is the square root of PDF_A Ohanian (talk) 20:26, 27 August 2017 (UTC)

First, terminology: you are not multiplying PDFs or taking their square roots; you are applying those operations to random variables, and wondering how to compute (or construct) the PDF of the result. For injective functions of a single variable this is easy. (It can be extended to simple non-injective functions by summing over the branches of their full inverses.) --Tardis (talk) 04:16, 28 August 2017 (UTC)

Using the Iverson bracket notation $[true]=1$ and $[false]=0$, if the probability density function of X is f and the probability density function of Y is g:

$\Pr(a<X<b)=\int _{a}^{b}f(x)dx=\int [a<x<b]f(x)dx$

$\Pr(a<Y<b)=\int [a<y<b]g(y)dy$

then obviously

$\Pr(a<X+Y<b)=\int \int [a<x+y<b]f(x)g(y)dydx$

$\Pr(a<X-Y<b)=\int \int [a<x-y<b]f(x)g(y)dydx$

$\Pr(a<{\frac {X}{Y}}<b)=\int \int [a<{\frac {x}{y}}<b]f(x)g(y)dydx$

and

$\Pr(a<{\sqrt {X}}<b)=\int [a<{\sqrt {x}}<b]f(x)dx$

Bo Jacoby (talk) 01:57, 31 August 2017 (UTC).