Suppose that your bus arrives at your bus stop uniformly between 9:05am and 9:10am. LetX= timeyou wait for the bus. Thus we have thatX∼Unif(5,10).(a) Give the PDF and CDF forX.(b) What is the expected time that the bus will arrive?(c) Suppose you slept in a little and can make it to the bus stop at 9:07. What is the probability thatyou will have missed the bus?

(a) Shown below

(b) The expected time it takes for the bus to arrive is 7.5 minutes.

(c) The probability that the bus was missed if I arrived at the bus stop at 9:07 AM is 0.40.

Step-by-step explanation:

Let X = waiting time for the bus.

It is provided that the bus arrives at the bus stop uniformly between 9:05 AM and 9:10 AM.

The random variable X is uniformly distributed with parameters a = 5 and b = 10.

The probability density function of a Uniform distribution is:

(a)

The probability density function of the random variable X is:

The cumulative distribution function of the random variable X is:

(b)

The expected value of Uniform distribution is:

Compute the expected value of X as follows:

Thus, the expected time it takes for the bus to arrive is 7.5 minutes.

(c)

If I reach the bus stop at 9:07 AM and I have missed the bus, then this implies that the bus arrived between 9:05 AM and 9:07 AM.

Compute the probability that the bus arrived between 9:05 AM and 9:07 AM as follows:

Thus, the probability that the bus was missed if I arrived at the bus stop at 9:07 AM is 0.40.

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step-by-step explanation: