Question

Question Eight:

i) Show using convolutions or moment generating functions that ;

(a) if ? and ? are independent random variables and ? has a ?_n  distribution and ? has a ?_m  distribution, then ? + ? has a ?_(m+n) distribution(These are Chi-distributions).

(b) If ?~???(?) and ?~???(?) are independent random variables, then ? + ? has a ???(? + ?) distribution.

ii) A company issues questionnaires to clients to obtain feedback on the clarity of their brochure. It is thought that 5% of clients do not find the brochure helpful. Calculate the approximate probability that in a sample of 1,000 responses, the number, ?, of clients who do not find the brochure helpful satisfies 40 < ? < 70.

Answer 1

here we will use if X and Y are independent then MGF of tgeir sum will be MGF of their product. the process is as

here for part 2, we will apply

given n=1000 and probability that client will be unsatisfied p=0.05, therefore number of clients(N) which will be unsatisfied be a random variable N and will follow binomial distribution with parameter n, p. but n is very large and p is very small therefore we will apply binomial approximation to poisson such as

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