Question

A bus arrives at a stop every 15 minutes exactly, in a very consistent way, very easily drawn. A passenger is not aware of the schedule, and arrives randomly at the stop. Let X represent the number of minutes they wait for the bus to arrive. What type of random variable is X, if the passenger arrives completely randomly at the stop?

Circle the correct answer:

Discrete Normal Uniform

Sketch a picture for X based upon your answer to part (a) and the information given in the problem. (Be sure to label the X and Y axes with numbers for your sketch for full credit.)

Determine the probability that X is greater than 9. Draw a picture and shade the area you use to calculate your answer.

Determine the probability that X is between 6 and 13 minutes. Draw a picture and shade the area you use to calculate your answer.

Answer 1

X is the number of minutes a passenger wait for the bus to arrive.

Since a bus arrives at every 15 minutes, the waiting time can be any time between 0 minutes and 15 minutes.

So X is a continuous random variable and waiting time is any time between 0 minutes and 15 minutes means X is uniformly distributed between 0 and 15.

So X is a continuous random variable uniformly distributed between 0 and 15.

So we can write the probability density function as

The graph of this density function will be a straight line parallel to X axis where X varies between 0 and 15 and y is 1/15.

(b)

Now,

The red color shaded area in rectangle is the required probability.

(c)

The red color shaded area in rectangle is the required probability.