# poisson process

• Dec 13th 2010, 02:13 AM
Beaky
poisson process
Let {N(t) : t >=0} be a Poisson process of rate 1 and let $\displaystyle T_{1} < T_{2} < ...$ denote the times of the points. Derive the pdf of $\displaystyle Y=T_{2}/T_{4}$

So I've been trying to work with the distribution function. I have:

$\displaystyle P(T_{2}/T_{4} > x) = P(T_{2} > xT_{4}) = P(N(xT_{4}) \in \{0,1\})$

I get stuck there though and can't find a similar example.
• Dec 13th 2010, 05:29 AM
Focus
Quote:

Originally Posted by Beaky
Let {N(t) : t >=0} be a Poisson process of rate 1 and let $\displaystyle T_{1} < T_{2} < ...$ denote the times of the points. Derive the pdf of $\displaystyle Y=T_{2}/T_{1}$

So I've been trying to work with the distribution function. I have:

$\displaystyle P(T_{2}/T_{4} > x) = P(T_{2} > xT_{4}) = P(N(xT_{4}) \in \{0,1\})$

I get stuck there though and can't find a similar example.

It may help you to prove that $\displaystyle T_n -T_{n-1}$ are i.i.d. exponentials (where T_0=0). Now $\displaystyle Y=\frac{T_2-T_1}{T_1}+1$ where both $\displaystyle T_2-T_1$ and $\displaystyle T_1$ are independent exponentials. Now use the law of total probability and condition one of them.
• Dec 13th 2010, 10:08 AM
Beaky
Sorry, typo in original post. It's $\displaystyle Y=T_{2}/T_{4}$. I'll try to use your advice anyways though.
• Dec 13th 2010, 11:45 AM
Beaky
I get to $\displaystyle P(\frac{T_{2}}{T_{4}}\le x)=P(\frac{T_{4}-T_{2}}{T_{2}}\ge \frac{1}{x}-1)$

But I get stuck here. I can't figure out how to apply conditioning and the law of total probability since the variables are continuous.
• Dec 13th 2010, 02:48 PM
Focus
Quote:

Originally Posted by Beaky
I get to $\displaystyle P(\frac{T_{2}}{T_{4}}\le x)=P(\frac{T_{4}-T_{2}}{T_{2}}\ge \frac{1}{x}-1)$

But I get stuck here. I can't figure out how to apply conditioning and the law of total probability since the variables are continuous.

I would suggest getting T_2-T_1 as it is exponential. The law of total probability says (for example)
$\displaystyle P(X+Y \in A)=\int P(X+y \in A)P(Y \in dy).$

So suppose that X=T_2-T_1, then X is exponential with parameter one and
$\displaystyle P(Y \leq y)=\int P(T_4\geq x/y)e^{-x} dx$
now you should try to figure out the law of T_4.