Finding the Distribution of two dependent RV's

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Hello,

My question is on 1.26 in the tut pack. We are given that K is the integer part of T, and I am assuming we cannot use convolution method in this instance to find the joint distribution. May you please assist me in working out the joint distribution.

Thanks

answered Mar 25 by (1,490 points)
selected Mar 26 by Rowan

Hi Kuwala

Firstly we need to find $$P[K=k, S\le s]$$.

We know that $$S=T-K$$ and the expression $$\{K=k\}$$ is equivalent to $$\{k\le T \lt k+1\}$$, hence:

$$P[K=k,S\le s] = P[k\le T \lt k+1,T-K\le s]$$

$$= P[k\le T \lt k+1, T \le k+s]$$

$$= P[k\le T\le k+s]$$ since we know $$s \varepsilon [0,1)$$.

Now we know that T is exponentially distributed so we can easily find the $$P[K=k, S \le s]$$ using integration. I will leave that for you to do.

We end up finding $$P[K=k, S\le s]$$ = $$e^{-\lambda k}$$ $$(1-e^{-\lambda s})$$ that is for any nonnegative integer k and $$s \varepsilon [0,1)$$.

The joint cumulative distribution of K and S can now be found.

$$F(k,s)$$ = $$P[K\le k, S\le s]$$

= $$\sum_{r=0}^k$$$$e^{-\lambda r}$$ $$(1-e^{-\lambda s})$$

I will leave the rest for you to do.

commented Mar 29 by (240 points)

Thank you so much Joshua