Question

NOTE :- Please find the probabilities as functions of cost

A service facility charges a $20 fixed fee plus $25 per hour of service up to 6 hours, and no additional fee is charged for a service visit exceeding 6 hours. Suppose that the service time τ again ranges from 0 to 10 hours, but now the probability density is twice as large during the middle 6 hours [2, 8] than during the outer 4 hours [0, 2] and [8, 10]. (Note as before that τ is a continuous random variable.) Let X represent the cost of service in the facility.

We would like to set up the probability density function (PDF) and the cumulative distribution function (CDF), then use them to analyze service fees.

Design Specifications

1. Sketch the probability density as a function of time. Be quantitative, and pay attention to units. Compute the probability that the service is greater than 6 hours. Then, sketch the probability density as a function of cost. Again, be quantitative and pay attention to units

2. Use the probability density function to set up the cumulative probability, also as a function of X.

Answer 1

Let be the service time. . Let k be the probability density of for and . Then the probability density for is .

Let be the probability density of . Then

Let be the cost of service. Then

The minimum value of is 2

The maximum value of is

Let be the probability density of and let be the cumulative distribution function of .

,

So for

, . So for

=

,since 170 is the maximum value of .

The cumulative distribution of (measured in dollars) is

The probability density function of is

The distribution has jump at

The value of the jump is

The probability density function as function of cost is given above where the cost is $x.

The cumulative distribution function is given above where the cost is .

Probability time greater than 6 hours is