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Probability Generating Function

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The probability generating function is a power series that gives us the probabilities associated with a discrete random variable.

Definition:

Let X be a discrete random variable that takes non-negative values 0, 1, 2,….etc. Let p_i=P(X=i) denote the probability that the random variable X takes the value ‘i’. Then the probability generating function (p.g.f) is a power series in ‘s’ (denoted P(s) ) defined as,

P(s) = E[s^x] = \sum_{x=0}^{\infty} s^xp_x

If we differentiate the above power series term by term ‘n’ times we get,

\frac{d^n}{ds^n}P(s) = n!p_n + (n+1).n\ldots.3.2 p_{n+1}s + \ldots

Substituing s=0 on both sides and rearranging terms we get,

p_n = P(X=n) = \frac{1}{n!}\frac{d^n P(s)}{ds^n} |_{s=0}

Thus we see that we can obtain the probablilities associated with a random variable by differentiating the probability geberating function.

Examples of Probability Generating Functions:

1. The probability generating function for the Poisson distribution with parameter \lambda is given as:

P(s) = e^{-\lambda}e^{s\lambda}

2. The probability generating function for the geometric distribution with parameter p is given as:

P(s) = \frac{p}{1-qs}

3. The probability generating function for the binomial distribution with parameters n and p is given as:

P(s) = (q+ps)^n

4. The probability generating function for the negative binomial distribution with parameters r andp is given as:

P(s) = (\frac{p}{1-qs})^r

Properties of P.G.F:

  1. Let X and Y be two independetn discrete random variables. Then the p.g.f of X+Y is equal to the product of their respective p.g.f’s, that is, P_{X+Y}(s) = P_X(s)P_Y(s)
  2. If the p.g.f is differentiated ‘r’ number of times, then on substituting s=1, we obtain the rth factorial moment.
  3. If we substitute s=e^t in the p.g.f then we obtain the moment generating function of our random variable.

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