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October 28th, 2009, 08:23 PM
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| |  Independent Geometric Random Variables
Let W1 and W2 be independent geometric random variables with parameters p1 and p2. Find
a) P(W1 = W2)
b) P(W1 < W2)
c) P(W1 > W2)
d) the distribution of min (W1 , W2)
e) the distribution of max (W1, W2)
I'm really stuck on this problem ... the geometric part is throwing me off. I would really appreciate any help! Thanks! |
October 28th, 2009, 09:54 PM
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Quote:
Originally Posted by bart203  Let W1 and W2 be independent geometric random variables with parameters p1 and p2. Find
a) P(W1 = W2)
b) P(W1 < W2)
c) P(W1 > W2)
d) the distribution of min (W1 , W2)
e) the distribution of max (W1, W2)
I'm really stuck on this problem ... the geometric part is throwing me off. I would really appreciate any help! Thanks! | Here's a start and something to think about :
a)  ![= \sum_{i = 1}^{+ \infty} (1 - p_1)^{i - 1} p_1 \cdot (1 - p_2)^{i-1} p_2 = \sum_{i = 1}^{+ \infty} [(1 - p_1)(1 - p_2)]^{i - 1} p_1 p_2](http://www.mathhelpforum.com/math-help/latex2/img/9ce9db430a7fa2c68367938909654e17-1.gif "= \sum_{i = 1}^{+ \infty} (1 - p_1)^{i - 1} p_1 \cdot (1 - p_2)^{i-1} p_2 = \sum_{i = 1}^{+ \infty} [(1 - p_1)(1 - p_2)]^{i - 1} p_1 p_2") ![= \frac{p_1 p_2}{(1 - p_1)(1 - p_2)} \sum_{i = 1}^{+ \infty} \left[ (1 - p_1) (1 - p_2)\right]^i](http://www.mathhelpforum.com/math-help/latex2/img/a341fa034c14c991ec20914c3e002836-1.gif "= \frac{p_1 p_2}{(1 - p_1)(1 - p_2)} \sum_{i = 1}^{+ \infty} \left[ (1 - p_1) (1 - p_2)\right]^i")
and now you have the sum of an infinite geometric series.
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Last edited by mr fantastic; October 29th, 2009 at 12:35 AM.
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October 28th, 2009, 11:18 PM
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Quote:
Originally Posted by mr fantastic Here's a start and something to think about :
a)  ![= \frac{p^2}{(1 - p)^2} \sum_{i = 1}^{+ \infty} \left[ (1 - p)^2\right]^i](http://www.mathhelpforum.com/math-help/latex2/img/29330f1c92abb1c9091ce1cde4983498-1.gif "= \frac{p^2}{(1 - p)^2} \sum_{i = 1}^{+ \infty} \left[ (1 - p)^2\right]^i")
and now you have the sum of an infinite geometric series. |
I may be mistaking, but does it not matter that the parameters are p1 and p2 respectively and not both p? | | The following users thank Juicy for this useful post: | | |
October 29th, 2009, 12:32 AM
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Quote:
Originally Posted by Juicy I may be mistaking, but does it not matter that the parameters are p1 and p2 respectively and not both p? | I will modify my post (the modification is simple, the basic argument is unchanged).
I will also note that the answer to c) is 1 - answer to a) - answer to b) .... (which saves some work).
__________________ There are two things you should never try to prove: the impossible and the obvious. The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low and achieving our mark. (Michelangelo Buonarroti)
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