Two independent random variables, X1 and X2, follow Negative Binomial distributions with parameters (p, k1) and (p, k2), respectively. Show that their sum X1+X2 follows a Negative Binomial distribution with parameters p and (k1+k2).

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M3
Two independent random variables, X1 and X2,
follow Negative Binomial distributions with
parameters (p, k1) and (p, k2), respectively. Show
that their sum X1+X2 follows a Negative Binomial
distribution with parameters p and (k1+k2).
The daily number of claims for a household
insurance portfolio has a Negative Binomial
distribution with p=0.3 and k=2.5. Assuming that
the claim numbers are independent from one day
to the other for this portfolio, calculate the
following quantities.
(i) Probability of 6 claims in four days.
(ii) Expected number of claims in a six day period.
(iii) Given that there were 4 claims in the first 3
days, the probability of 3 more claims in the next
two days.
Transcribed Image Text:Two independent random variables, X1 and X2, follow Negative Binomial distributions with parameters (p, k1) and (p, k2), respectively. Show that their sum X1+X2 follows a Negative Binomial distribution with parameters p and (k1+k2). The daily number of claims for a household insurance portfolio has a Negative Binomial distribution with p=0.3 and k=2.5. Assuming that the claim numbers are independent from one day to the other for this portfolio, calculate the following quantities. (i) Probability of 6 claims in four days. (ii) Expected number of claims in a six day period. (iii) Given that there were 4 claims in the first 3 days, the probability of 3 more claims in the next two days.
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