Manipulating PGF for Probability Calculation

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In summary, the conversation is about finding P(X=i) from the given PGF Gx(s)= (4-s)/3(4-3s) and manipulating it to obtain the sum to infinity of a geometric series. However, the term (4-s) remains in P(X=i) and the person is looking for hints on how to rearrange the equation to eliminate the s in the numerator.
  • #1
jackbauer
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Hi people, I was wondering if anyone could give me a hint with this problem.
A RV X has PGF Gx(s)= (4-s)/3(4-3s)

I need to get the P(X=i) from here. I know i can manipulate the above to get the sum to infinity of a geometric series, but i end up with the term
(4-s) still in the P(X=i) which i don't think can be right. Anybody got any hints on how to rearrange the above so as to eliminate the s in the numerator so I can obtain P(X=i)? Thanks a lot, Jack
 
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  • #2
multiply out the (4-s) and the infinite series.
 

FAQ: Manipulating PGF for Probability Calculation

What does "P(X=i) from given PGF" mean?

"P(X=i) from given PGF" refers to the probability of a random variable X taking the value i, given the probability generating function (PGF) of X. The PGF is a mathematical function that describes the distribution of a discrete random variable.

How do you calculate P(X=i) from a given PGF?

To calculate P(X=i) from a given PGF, you can use the formula P(X=i) = pi/i!, where pi is the coefficient of the term xi in the PGF. This formula applies for discrete random variables with non-negative integer values.

Can the PGF be used to calculate probabilities for continuous random variables?

No, the PGF is only applicable for discrete random variables. For continuous random variables, the probability density function (PDF) or cumulative distribution function (CDF) should be used to calculate probabilities.

What information can the PGF provide about a random variable?

The PGF can provide information about the distribution of a random variable, such as the mean, variance, and higher moments. It can also be used to calculate probabilities and generate random samples from the distribution.

Are there any limitations to using the PGF to calculate probabilities?

Yes, the PGF can only be used for discrete random variables with non-negative integer values. Additionally, it may not be possible to find a closed-form expression for the PGF for all distributions, in which case numerical methods may need to be used to approximate probabilities.

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