What is the distribution of $\frac{X_1}{X_2}$ if both $X_1$ and $X_2$ are the Poisson Processes with parameters $\lambda_{1}$ and $\lambda_{2}$ respectively? Please list the properties used too. Thanks !
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1Too little information... – rlartiga Apr 25 '14 at 17:03
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Is conditional or division? – rlartiga Apr 25 '14 at 17:05
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1The distribution of the quotient of two Poisson random variables is obviously not well-defined--there is a nonzero probability that $X_1/X_2$ will either be infinite, or indeterminate of the form $0/0$. – heropup Apr 25 '14 at 17:09
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Its Division.. Thanks – user2778430 Apr 25 '14 at 17:09
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I am sorry... X1 and X2 are Poisson Processes not just RV – user2778430 Apr 25 '14 at 17:13
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Try changing $X1/X2$ for $X_1/(X_2+1)$ for make more sense. Anyway the result is not going to be nice, – rlartiga Apr 25 '14 at 17:43
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$$P\left(\frac{N_1(t)}{N_2(t)}\leq x|N_2(t) \geq 1\right)=P\left(N_1(t)\leq N_2(t)x |N_2(t) \geq 1\right)=$$ $$\sum_{i=1}^\infty P\left(N_1(t)\leq ix\right)P(N_2(t)=i |N_2(t)\geq 1)=\sum_{i=1}^\infty \sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)P(N_2(t)=i|N_2(t)\geq 1)$$ $$=\sum_{i=1}^\infty\sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)\frac{P(N_2(t)=i)}{P(N_2(t)\geq 1)}$$ $$=\frac{1}{1-e^{-\lambda_2t}}\sum_{i=1}^\infty\sum_{j=0}^{[ix]} P\left(N_1(t)=j\right)P(N_2(t)=i)$$
rlartiga
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