The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150. Part (a)

Question

For each probability and percentile problem, draw the picture.

The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.

Part (a)

In words, define the Random Variable X.

1. the cost, in dollars, of maintenance for a car during its first year
2. the number of times a car needs maintenance during its first year    
3. the total cost, in dollars, of maintenance for a car during its lifetime
4. the cost, in dollars, of a new car

Part (b)

1. Give the distribution of X. (Enter the numerical value as a fraction.)

X ~  _____ ( _____)

  

Part (c)

Enter an exact number as an integer, fraction, or decimal.

1. μ = ____

Part (d)

Enter an exact number as an integer, fraction, or decimal.

1. σ = ______

Part (e)

1. Draw a graph of the probability distribution.

Part (f)

Find the probability that a car required over $320 for maintenance during its first year. (Round your answer to four decimal places.) _____

Answer 1

(a) will be the first

(b) Exp (1/150)

(c) Mean = 1/150

(d) Standard deviation = 1/150

(e) The pdf, f(x) = λe^-λx, where λ is 1/mean (1/150). So when x=0, f(x) = 1/150 and as x gets larger and larger f(x) tends to zero from above. 1/150 = 0.0067, so when x=0 f(x) = 0.0067 so must be first graph.

(f) The cdf is F(x) = 1 - e^-λx

So we need to find F(320) = 1 - e^-320/150 = 0.88156

This is the probability that the cost is <= 320

So to get the required answer we take this from 1 giving 0.11844 , 0.1184 to 4 decimal places