Question

(2) A professor has a total of 400 students in her calculus classes. On a given day, each student has a 1/20 chance of coming to her office hours. Let X be the number of students who come to office hours on a given day.

(a) Find the pmf of X.

(b) Repeat part (a) by approximating the pmf of X with a Poisson pmf.

(c) The professor’s office accommodates only 10 students. What is the probability that 10 or less students come to office hour? Provide the exact formula using (a) and an approximation using (b).

(d) What is the probability that the number of students who come to office hour is exactly equal to the expected value of X? Use a calculator, if necessary, to write down your answer as one number.

(e) Find the probability in the previous part using the Poisson approximation. Use a calculator, if necessary, to write down your answer as one number.

Answer 1

total number of students = n = 400

p = probability of students coming to her office hours = 1/20 = 0.05

X = be the number of students who come to office hours on a given day.

So the range of X from 0 to 400

The distribution of a random variable X is binomial ( n = 400, p = 0.05)

b) Since n = 400 is very large and p = 0.05 is very small

therefore we can used Poisson approximation to the Binomial

c) We want to find P(X <= 10)

Using excel:

Exact probability:

P(X <= 10) = "=BINOMDIST(10,400,0.05,1)" = 0.009399

Approximate probability using Poisson distribution

P(X <= 10) = "=POISSON(10,20,1)" = 0.010812

d) E(X) = n *p = 400/20 = 20

P(X = 20) = "=BINOMDIST(20,400,0.05,0)" = 0.091142

e) Using Poisson distribution:

P(X = 20) = "=POISSON(20,20,0)" = 0.088835