Wikipedia:Reference desk/Archives/Mathematics/2017 August 27
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August 27
editWikipedia article about adding subtracting dividing and multiplying two PDF
editA 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 and , if the probability density function of X is f and the probability density function of Y is g:
![{\displaystyle [true]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feccd10a2d114432776ca5586940b3932f98317d)
![{\displaystyle [false]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/459e17c637f5c830cf1c8e7846622f62af74c33c)
![{\displaystyle \Pr(a<X<b)=\int _{a}^{b}f(x)dx=\int [a<x<b]f(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b50fb5311643208db986e0317726ebeb5f21825c)
![{\displaystyle \Pr(a<Y<b)=\int [a<y<b]g(y)dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aae64a50d324cbeb43bc9fb58cc03f384a707c1)
then obviously
![{\displaystyle \Pr(a<X+Y<b)=\int \int [a<x+y<b]f(x)g(y)dydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81bb0d46bbc90b28695a49c5d699edf8149137c4)
![{\displaystyle \Pr(a<X-Y<b)=\int \int [a<x-y<b]f(x)g(y)dydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcd91acd09996ad209f55d4fc1bc75b557d134e)
![{\displaystyle \Pr(a<{\frac {X}{Y}}<b)=\int \int [a<{\frac {x}{y}}<b]f(x)g(y)dydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74533328707756a9a7ed19216cc38a0174865eac)
and
![{\displaystyle \Pr(a<{\sqrt {X}}<b)=\int [a<{\sqrt {x}}<b]f(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca15f28e21015b9f2c248625434137c599ca2970)