Question

Question 1.

Recall your consultancy work for Drovandi Marketing and Networks (DMN) from your last PST. In the final phase of your current project, MXB101, you are required to delve into the minds of customers. DMN has taken a large survey of customers buying a particular product. From this survey, they know that:

• 25% of customers are under 25.
• 60% of customers are between 25 and 50.
• 15% of customers are over 50.

DMN estimates that 20% of the population in the local area have bought this product. DMN has surveyed locals who haven’t purchased the product and estimates that:

• 70% of non-customers are under 25.
• 10% of non-customers are between 25 and 50.
• 20% of non-customers are over 50.

What is the probability that a person under 25 will buy this product?

Question 2.

Outpatient clinics are hospital clinics that have appointments for patients who visit the hospital without being admitted. At one particular clinic, it is known that the distribution of appointment times (in hours), X, is described by the cumulative distribution function

                                                                                          F_X(x) = k − e^−x2,         x ≥ 0.

1. Find k such that F_X(x) is a valid cdf.
2. What is the probability that a patient has an appointment longer than one hour?
3. (Original phrasing) One particular patient must complete a 30 minute exercise program during their appointment. Show that the probability this patient’s appointment will then exceed one hour in total is e−3/4 ≈ 0.47.

3. (Modified phrasing) You arrived early for an appointment and watched the previous patient enter for their appointment. You look at your watch and see that the previous patients’ appointment has been running for 30 minutes and has not yet finished. Show that the probability that the previous patients’ appointment will take longer than 1 hour is

e−3/4 ≈ 0.47.

4. Write down an expression for the average appointment time, E(X).

Note: You do not have to evaluate the integral you write down.

Question 3.

The number of business days in a week, X, that the outpatient clinic is over capacity is a random variable with the following probability mass function.

x	0	1	2	3	4	5
Pr(X = x)	0.04	0.23	a	0.19	0.12	b

Find the value of a and b such that E(X²) = 5.75.

can you help me answer these questions?

Answer 1