# Thread: distribution function

1. ## distribution function

Hi can you help me to solve the following problem?. thank you

Let X1 and X2 denote a random sample of size 2 from a distribution with p.d.f. (densitity function) f(x)=1 , 0<x<1, zero elsewhere. Find the distribution function and p.d.f. of Y=X1/X2

2. Originally Posted by user
Hi can you help me to solve the following problem?. thank you

Let X1 and X2 denote a random sample of size 2 from a distribution with p.d.f. (densitity function) f(x)=1 , 0<x<1, zero elsewhere. Find the distribution function and p.d.f. of Y=X1/X2
Read these:

Ratio Distribution -- from Wolfram MathWorld

Ratio distribution - Wikipedia, the free encyclopedia

Quotient of two random variables

3. ## Again

Hi can you help me to solve the following problem?. But without stochastics theory. I must to use theory of probability and transformations. Thank you

Let X1 and X2 denote a random sample of size 2 from a distribution with p.d.f. (densitity function) f(x)=1 , 0<x<1, zero elsewhere. Find the distribution function and p.d.f. of Y=X1/X2

4. Originally Posted by user
Hi can you help me to solve the following problem?. But without stochastics theory. I must to use theory of probability and transformations. Thank you

Let X1 and X2 denote a random sample of size 2 from a distribution with p.d.f. (densitity function) f(x)=1 , 0<x<1, zero elsewhere. Find the distribution function and p.d.f. of Y=X1/X2
If you read the links I gave you will know that the distribution of Y/X where X and Y are continuous independent random variables with pdf's f(x) and g(y) respectively is given by

$\displaystyle h(u) = \int_{-\infty}^{+\infty} |y| \, f(uy) \, g(y) \, dy$. This is easily proved using the 'Change of variable (transformation)' theorem.

For your problem, note that f(uy) = 1 if $\displaystyle 0 \leq 0 uy \leq 1 \Rightarrow 0 \leq y \leq \frac{1}{u}$ and zero otherwise.

Then:

Case 1: h(u) = 0 for u < 0.

Case 2: $\displaystyle h(u) = \int_0^1 y \, dy$ for $\displaystyle 0 \leq u \leq 1$.

Case 3: $\displaystyle h(u) = \int_0^{1/y} y \, dy$ for $\displaystyle u > 1$.