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Old 11-14-2017, 07:00 AM
Nagsinde2002 Nagsinde2002 is offline
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Default Joint PMFs

Hello. I have been struggling on this question.

Let X and Y be independent Geo(1/3) random variables. Let V =min(X,Y) and let W be defined as follows:
W = 0 if X< Y,
W=1 if X=Y
W=2 if X > Y.

Let p{V,W}(v,w) be the joint probability mass function of V and W. Compute p{V,W}(2,0).

I know that I have to make a pmf of V and W in terms of X and Y, but I am stuck on how to do so. Also, how would I use the geometric random variable pmf into this problem? If someone can get me started on this problem, that would be appreciated.
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Old 11-14-2017, 11:09 AM
ARodOmaha ARodOmaha is offline
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Quote:
Originally Posted by Nagsinde2002 View Post
Hello. I have been struggling on this question.

Let X and Y be independent Geo(1/3) random variables. Let V =min(X,Y) and let W be defined as follows:
W = 0 if X< Y,
W=1 if X=Y
W=2 if X > Y.

Let p{V,W}(v,w) be the joint probability mass function of V and W. Compute p{V,W}(2,0).

I know that I have to make a pmf of V and W in terms of X and Y, but I am stuck on how to do so. Also, how would I use the geometric random variable pmf into this problem? If someone can get me started on this problem, that would be appreciated.
I'll get you started and you can finish. Deciphering the last statement and working backwards, we need the probability that V=2 and W=0. If W=0 then we know X<Y. Also, since V=Min[X, Y], then we know that X=2 and Y>2.

To find the probability that X=2 and Y>2, that is the Pr(X=2)*Pr(Y>2). For Pr(X=2), plug into the Geometric distribution: (1-1/3)^(2-1) * (1/3) = 2/9. Now calculate Pr(Y>2) as the complement 1-P(Y<=2)....there are three options to consider. Multiply that result with 2/9.
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Old 11-14-2017, 02:43 PM
Nagsinde2002 Nagsinde2002 is offline
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Thanks for your help. I got the answer correct, which should be 8/81.
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Old 11-20-2017, 11:12 AM
Z3ta Z3ta is offline
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The problem alone doesn't tell you whether geometric is counting failures or trials here. Your answer of is correct if you're counting trials. If you're counting failures it would be .

Also just for fun and in case you find this useful (all true whether counting failures or trials):
The PMF of is


is Geometric with parameter

In general if and are independent and geometric with parameters and respectively, then is geometric with parameter .

Last edited by Z3ta; 11-20-2017 at 11:28 AM..
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