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Finding the Distribution of two dependent RV's

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asked Mar 24 in BUS 3018F - Models by Kuwala (240 points)

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

1 Answer

+3 votes
answered Mar 25 by joshua_wort (1,490 points)
selected Mar 26 by Rowan
 
Best answer

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 Kuwala (240 points)

Thank you so much Joshua

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