Question

4. (a) Susan tries to exercise at ”Pure Fit” Gym each day of the week, except on the weekends (Saturdays and Sundays). Susan is able to exercise, on average, on 75% of the weekdays (Monday to Friday).

1. Find the expected value and the standard deviation of the number of days she exercises in a given week.    [2 marks]
2. Given that Susan exercises on Monday, find the probability that she will exercise at least 3 days in the rest of the week. [3 marks]
3. Find the probability that in a period of four weeks, Susan exercises 3 or less days in only two of the four weeks.      [3 marks]

2. A car repair shop uses a particular spare part at an average rate of 6 per week. Find the probability that:
   1. at least 6 are used in a particular week.                                                            [2 marks]
   2. exactly 18 are used in a 3-week period.                                                             [3 marks]
   3. exactly 6 are used in each of 3 successive weeks.

PLZ ans 4B

Answer 1

4.

(b)

These 3 problems can be solved using Poisson distribution with the following property.

If , then .

(i)

Suppose, random variable X denotes number of spare parts used in a week.

Required probability is given by

[Using R-code 'ppois(5,6)']

(ii)

Suppose, random variable Y denotes number of spare parts used in a 3 week period.

Required probability is given by

  [Using R-code 'dpois(18,18)']

(iii)

Number of spare parts used in a week is independent of number of spare parts used in other weeks.

We have,

Probability of using exactly 6 spare parts in a week is given by

  [Using R-code 'dpois(6,6)']

So, probability of using exactly 6 spare parts in each of 3 successive weeks