# Get a Probability Distribution Function of a Multivariate Function

I have a function $a = x/y$ where $x$ can be represented with a uniform distribution $[21,26]$ and $y$ can be represented with a normal distribution with a certain mean and stdev. $x$ and $y$ are independent.

Now I would like to combine the two distributions to get a pdf for $a$. How can I do this? I am somehow stuck with joint probability distributions and marginals and I don't find a way to start!

Thanks for your help, I really appreciate!

• What is "univariate distribution [21,26]"? May be, it is the uniform distribution? Are $x$ and $y$ independent? – NCh Jun 7 '17 at 15:15
• en.wikipedia.org/wiki/Ratio_distribution#Derivation – NCh Jun 7 '17 at 15:20
• @NCh Yes, uniform distribution, sorry! Thank you for the link!Will this principle also work if e.g. $a=b*x/(d+y^(1/3))$? – Matt Jun 7 '17 at 15:59
• To use formula from wiki in this case you should previously find the pdf's of numerator and denominator separately. – NCh Jun 7 '17 at 16:36

$$f_{X}(x) = \mathbb{I}(x\in[21,26])$$

and

$$f_{Y}(y) = \frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}y^2)$$

$X$ is independent of $Y$ $\Rightarrow f_{X,Y}(x,y) = f_X(x)f_Y(y)$

Now, let $U = Y$ and $A=\frac{X}{Y}$. Therefore, $Y = U$ and $X = AU$ and

$$J = \begin{pmatrix} A & U \\ 1 & 0\end{pmatrix} \Rightarrow |detJ| = U$$

So,

$$f_{U,A}(u,a) = f_{X,Y}(au, u) . u = \mathbb{I}(au \in [21,26])\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}u^2).u$$

Finally,

$$f_{A}(a) = \int_{u:\, au \in [21,26]}f_{U,A}(u,a)du = \begin{cases} \frac{1}{\sqrt{2\pi}}(\exp(-\frac{21}{a})-\exp(-\frac{26}{a})) \text{ if } a > 0 \\ \frac{1}{\sqrt{2\pi}}(\exp(\frac{26}{a})-\exp(\frac{21}{a})) \text{ if } a < 0 \end{cases}$$