Question

Calculate ?(? < 8) if: (i) ? is the number of distinctions reported in a year by 20 Colleges. Each College produces distinctions at the rate of 0.2 per year independently of the other Colleges. (ii) ? is the number of claims examined up to and including the fourth claim that exceeds K20,000. The probability that any claim received exceeds K20,000 is 0.3 independently of any other claim. (iii) ? is the number of deaths amongst a group of 500 TB patients. Each patient has a 0.01 probability of dying independently of any other patient. (iv) ? is the number of phone calls made before an agent makes the first sale. The probability that any phone call leads to a sale is 0.01 independently of any other call.

Answer 1

i) The number of distinctions reported in a year by 20 Colleges follows Binomial distribution with

The PMF of is  .

The probabilty,

Use R to find the probability,

pbinom(7,20,0.2)
[1] 0.9678573

ii) is the number of claims examined up to and including the fourth claim that exceeds $20,000. So we need 3 claims that exceeds $20000 in tosses, and the 4-th such claim occurs in the -the claim . So the distribution (PMF) of is

So has hypergeometric distribution.

Use R to compute the probability,

> pnbinom(3,size = 4, prob = 0.3)
[1] 0.126036

c) ? is the number of deaths amongst a group of 500 TB patients. . We can use Poisson approximation to Binomial, with .

The Poisson PMF is

The probability,

R command is

> ppois(7,5)
[1] 0.8666283

d)? is the number of phone calls made before an agent makes the first sale. The probability that any phone call leads to a sale is 0.01 independently of any other call. Hence   has geometric distribution.

with .

The geometric PMF is

R command below.

> pgeom(6,0.01)
[1] 0.06793465