Fall 2011 Homework 9: Question 3
Consider independent Bernoulli trials with probability p of success for each. Let X be the number of failures incurred before getting a total of r successes.
(a) Determine what happens to the distribution of $\frac{p}{1-p}X$ as $p\rightarrow 0$, using MGFs; what is the PDF of the limiting distribution, and its name and parameters if it is one we have studied? Hint: start by finding the Geom(p) MGF. Then find the MGF of $\frac{p}{1-p}X$, and use the fact that if the MGFs of r.v.s $Y_{n}$ converge to the MGF of a r.v. Y, then the CDFs of the $Y_{n}$ converge to the CDF of Y.
(b) Explain intuitively why the result of (a) makes sense.
Solution: (a) This is the Gamma(r, 1) MGF for t < 1 (note also that the condition qe^(tp/q) < 1 is equivalent to t < (p-1)/p * log(1-p), which converges to the condition t < 1 since again by L'Hopital's Rule, -p/log(1-p) goes to 1). Thus, the scaled Negative Binomial Xp/(1-p) converges to Gamma(r, 1) in distribution as p goes to 0. (b) Gamma is the continuous analogue of the Negative Binomial, just as the Exponential is the continuous analogue of the Geometric (as discussed in class and seen on the HW 5 problem about The Winds of Winter). See iTunes course for further detail.
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