Problem: Consider one way of modeling a problem. Say you have a device an in any given year there is about a 1 in 6 chance that the device will fail. A question is, "On average, how long will it be before such a device fails?" This sort of problem is what will be modeled and analyzed here.

The problem above can be modeled as follows: Roll a normal fair six-sided die. Rolling a 1 will count as "fail" anything else will count as "not-fail". The random variable we will use will be

X = # of rolls required before a 1 is rolled = "time until failure"

This is a random variable, so in each experiment (for each device) this will produce a value. For example, if we roll, then X = 5 (we count the final roll).

Part 1 (Empirical Analysis): Roll a die, or simulate using Excel, python, or some other option, e.g., Random.org, 20 repetitions of obtaining a value for X, that is, roll until a 1 is rolled, record the result and repeat 20 times. For example, if the rolls are

, then record 4 since 4 rolls were required.

Record your rolls and counts.

Compute the mean and standard deviation of your values for X.

Given the modeling problem we started with interpret the mean and standard deviation you found in terms of how many years are expected before the device fails.

Excel might be used for this part as it makes the calculations, recording of data, etc., very simple, however, it is not required.

Part 2 (Theoretical Analysis): Compute the expected value, variance, and standard deviation of the random variable X. See the additional notes for discussion on these computations, in brief:

= the probability of rolling something other than a 1 )-times, then rolling a 1 on the i^th throw. Make a table for the first 10 values:

Use this to compute and and then use these to get a rough approximation of and

(Expected Value)

(Variance)

(Standard Deviation)

Repeat the second item of Part 1:Given the modeling problem we started with, interpret and in terms of how many years are expected before the device fails.

Hint: For computing these without directly manipulating the infinite summations that appear in the definitions of and , let be the event that a 1 is rolled on the first throw and be the complementary event, namely, that a 1 is not rolled on the first throw. It is clear that since what is rolled after the first roll is just like starting over. It is also clear that . The Law of Total Expectation (see notes) gives:

This makes it quite simple to find .

A similar "trick" can be used to find , here you will use .Again, the Law of Total Expectation gives:

and from this it is simple to compute .