Question

1. The maximum amount of time a guest at the Holiday Inn can wait for an elevator is 4 minutes. Assuming the wait time follows a uniform distribution:

1. How long should a guest expect to wait?

2. What is the standard deviation of wait times?

3. What is the probability that a guest waits exactly 2 minutes for the elevator?

4. What is the probability that a guest waits more than 3 minutes?

5. What is the probability that a guest waits less than 90 seconds?

6. What is the probability that a guest waits between 1 minute and 2:30 for the elevator?

Answer 1

This is a uniform distribution with

Since we know that
Probability density function of a uniform distribution is

This implies that
Cummulative density function of a uniform distribution is

a)Since we also know that
Mean of a uniform distribution is the average of its interval i.e.

Mean = 2.0
b) Also

Variance = 1.3333

Standard Deviation = 1.1547
c) For a continous probability distribution, probablity at a particular point is equal to zero
P(X=2) = 0
d) Pr(X>x) = 1- F(x)
Where x = 3

Pr(X>3.0) = 0.25
e) Pr(X<x) = F(x)
Where x = 1.5

Pr(X<1.5) = 0.375
f) Pr(x₁<X<x₂) = F(x₂) - F(x₁)
Where x1 = 1
x₂ = 2.5

Pr(1.0<X<2.5) = 0.375