Question

After graduating from Medical School, Dr. Jane opened a clinic. Because of her specialty, she anticipated spending with each patient about one hour on average with a standard deviation of 20 minutes. Neither to prolong her workday nor to keep it too short, she instructed her secretary thus: "Please schedule no more than 6 patients on each working day. Always finalize a day's appointments at least a week ahead of the day. Should there be only 1 or 2 patients scheduled on a day, cancel their appointments; reschedule them on the next day with at least three patients and an open slot. On these newly vacated days and on the ones with no patient scheduled at all, I will volunteer at the Community Relief Center."

Suppose that before any rescheduling is done, the number of patients, X, Dr. Jane is supposed to see takes values 0, 1, 2, ..., 6 with associated probabilities 0.05, 0.10, 0.10, 0.25, 0.25, 0.15, 0.10. What is the probability mass function of the number of patients, Y, Dr. Jane will see because of her instructions to the secretary? What proportion of workdays will she be volunteering at CRC instead? What are the mean and the standard deviation of the total amount of time she will be spending with patients on her clinic days？

Answer 1

Answer : -

In other words, for a discrete random variable X, the value of the Probability Mass Function P(x) is given as,

P(x)= P(X=x)

x	0	1	2	3	4	5	6
P(X=x)	0.05	0.10	0.10	0.25	0.25	0.15	0.10

This is the PMF (probability mass function ).

Conditions of PMF :

1) 0 <= p(x) <= 1 for all x and ∫ p(x) dx =1

2)

Now we want to find the mean and standard deviation.