# Probability: expected value using exponential distribution

Please assist in answering the following question. Please submit details of work including excel sheet used to arrive to the solution. Provide interpretation of results and describe conclusions.

1. Suppose that a car rental agency offers insurance for a week that will cost $10 per day. A minor fender bender will cost $1,500, while a major accident might cost $15,000 in repairs. Without the insurance, you would be personally liable for any damages. What should you do? Clearly, there are two decision alternatives: take the insurance or do not take the insurance.

The uncertain consequences, or events that might occur, are that you would not be involved in an accident, that you would be involved in a fender bender, or that you would be involved in a major accident. Assume that you researched insurance industry statistics and found out that the probability of major accident is 0.05%, and that the probability of a fender bender is 0.16%. What is the expected value decision? Would you choose this? Why or why not? What would be some alternate ways to evaluate risk?

2. Suppose that the service rate to a waiting line system is 10 customers per hour (exponentially distributed). Analyze how the average waiting time is expected to change as the arrival rate varies from two to ten customers per hour (exponentially distributed).

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#### Solution Summary

This solution is comprised of a detailed explanation of two probability questions using exponential distribution. This solution is comprised of calculations, formula and interpretation of results for both questions using the exponential distribution.

Probability: Standard Deviation, Exponential Distribution and Expected Values

(8) 1. A manufacturer of electronics products expects that 1% of units fail during the warranty period. A sample of 600 independent units is tracked for warranty performance.

a) What is the expected number of failures during the warranty period?

b) What is the probability that none fails during the warranty period?

c) What is the probability that more than three units fail during the warranty period?

d) What is the probability that the number of failures will be between 1, 2, 3 standard deviations?

(4) 2. A binary message is transmitted as a signal S, which is either ? 1 or + 1. The communication

channel corrupts the transmission with additive normal noise with = .1 and variance cr2 2.

The receiver concludes that the signal ? 1 was transmitted if the value received is < 0.

What is the probability of error?

(7) 5. A system consists of two components connected a) in series ; b) in parallel.

Suppose that i-th component has a lifetime that is exponentially distributed with ... = .01 and

.. = .02, and that components fail independently. Let X = the time at which the system fails.

Calculate the cdf F(x) = P(X ≤ x), the pdf of X and the expected value of X for cases a) and b).

Hint: use notation X = lifetime of i-th component, A = {X2 <x}, A = {X <x}, i = 1, 2.

(5) 6.* A stick of a length one is broken at random. Find the expected value of the difference between the larger part and the smaller part.