Probability: Infinite Convergent Series and Random Variables

AI Thread Summary
A user seeks to validate their probability mass function, expressed as Σ 12/(k+4)(k+3)(k+2), and confirms its convergence to 1 using Maple. However, they require a mathematical proof rather than relying on software validation. They initially attempted partial fraction decomposition but found it did not yield a telescoping series. Another participant points out that the decomposition can indeed be arranged to form a telescoping series, leading to the user's realization of the oversight. This exchange highlights the importance of mathematical rigor in proving the validity of probability functions.
ZellDincht100
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I have a random variable problem. I need to prove that my equation I came up with is a valid probability mass function.

In the problem, I came up with this for my probability mass function:

\Sigma 12/(k+4)(k+3)(k+2)

Maple says that this does in fact converge to 1, so it's valid; however...I can't use "Maple said so" as an answer.

My attempt was to break it up using partial fraction decomposition:
(6/(k+4)) - (12/(k+3)) + (6/(k+2))

I was hoping that this would be telescoping, but it is not. Does anyone have an idea on how I can prove that this converges to 1?
 
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Hi ZellDincht100! :smile:
ZellDincht100 said:
My attempt was to break it up using partial fraction decomposition:
(6/(k+4)) - (12/(k+3)) + (6/(k+2))

I was hoping that this would be telescoping, but it is not.

Yes it is …

[6/(k+4) - 6/(k+3)] - [6/(k+3) - 6/(k+2)] :wink:​
 
tiny-tim said:
Hi ZellDincht100! :smile:


Yes it is …

[6/(k+4) - 6/(k+3)] - [6/(k+3) - 6/(k+2)] :wink:​

Ahhhh I see! :D

Thanks! Dunno how I didn't see that before..
 

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