Probability: Infinite Convergent Series and Random Variables

In summary, the conversation is about proving the validity of an equation for a probability mass function. The equation is (6/(k+4)) - (12/(k+3)) + (6/(k+2)) and the speaker attempted to use partial fraction decomposition to prove convergence to 1, but it was not telescoping. Another person suggested a solution, which the speaker thanks them for.
  • #1
ZellDincht100
3
0
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:

[tex]\Sigma[/tex] [tex]12/(k+4)(k+3)(k+2)[/tex]

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:
([tex]6/(k+4)[/tex]) - ([tex]12/(k+3)[/tex]) + ([tex]6/(k+2)[/tex])

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?
 
Physics news on Phys.org
  • #2
Hi ZellDincht100! :smile:
ZellDincht100 said:
My attempt was to break it up using partial fraction decomposition:
([tex]6/(k+4)[/tex]) - ([tex]12/(k+3)[/tex]) + ([tex]6/(k+2)[/tex])

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:
 
  • #3
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..
 

1. What is an infinite convergent series?

An infinite convergent series is a sequence of numbers that approaches a finite limit as the number of terms increases. It is the sum of an infinite number of terms that have a well-defined pattern or formula.

2. How is probability related to infinite convergent series?

Infinite convergent series are used in probability to represent the likelihood of an event occurring. The sum of all possible outcomes in a probability distribution must equal 1, which can be achieved by representing the probabilities as an infinite convergent series.

3. What are random variables?

Random variables are numerical quantities that are determined by the outcome of a random event. They have a probability distribution associated with them, which describes the likelihood of different values occurring for that variable.

4. How are random variables used in probability?

Random variables are used to model and analyze random events or processes in probability. They allow for the quantification of the likelihood of different outcomes and can be used to calculate probabilities and expected values.

5. What is the significance of the central limit theorem?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will approximate a normal distribution. This is significant because it allows for the use of normal distribution properties in analyzing and making predictions about a wide range of random processes and events.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
30
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
865
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
820
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
388
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
781
Back
Top