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

= 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
 * $$\Pr(a<X+Y<b)=\int\int [a<x+y<b] f(x) g(y) dy dx$$
 * $$\Pr(a<X-Y<b)=\int\int [a<x-y<b] f(x) g(y) dy dx$$
 * $$\Pr(a<\frac X Y <b)=\int\int [a<\frac x y <b] f(x) g(y) dy dx$$

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

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