Grinstead-Snell-Probability · Problem 6.3.16

Sums of Random Variables

Mathematics · expected-value-variance-continuous

Problem

Assume that the lifetime of a diesel engine part is a random variable

X

with density

f_X

. When the part wears out, it is replaced by another with the same density. Let

N(t)

be the number of parts that are used in time

t

. We want to study the random variable

N(t)/t

. Since parts are replaced on the average every

E(X)

time units, we expect about

t/E(X)

parts to be used in time

t

. That is, we expect that

\lim_{t \to \infty} E\!\left(\frac{N(t)}{t}\right) = \frac{1}{E(X)}\,.

This result is correct but quite difficult to prove. Write a program that will allow you to specify the density

f_X

, and the time

t

, and simulate this experiment to find

N(t)/t

. Have your program repeat the experiment 500 times and plot a bar graph for the random outcomes of

N(t)/t

. From this data, estimate

E(N(t)/t)

and compare this with

1/E(X)

. In particular, do this for

t = 100

with the following two densities: (a)

f_X = e^{-t}

. (b)

f_X = te^{-t}