362 CHAPTER 6 Inferential Statistics
- You have determined that along a stretch of highway, you see on
average one dead animal on the road every 2.1 km. Assuming an
exponential distribution with this mean, what is the probability
that after seeing the last road kill you will drive 8 km before seeing
the next one. - (Harder question) Assume, as in the above problem that you see on
average one dead animal every 2.1 km along the above-mentioned
highway. What is the probability that you will drive at least 10
km before seeing the next two dead animals? (Hint: LetX 1 be
the distance required to spot the first roadkill, and let X 2 be the
distance required to spot the second roadkill. You’re trying to
computeP(X 1 +X 2 ≥10); try looking ahead to page 370.) - We can simulate the exponential distribution on a TI-series calcu-
lator, as follows. We wish to determine the transforming function
g such that when applied torand results in the exponential ran-
dom variableY with density functionfY(t) =λe−λt. Next, from
page 353 we see that, in fact
∫t
0
fY(x)dx=
∫t
0
λe−λxdx=g−^1 (t).That is to say, 1−e−λt=g−^1 (t).(a) Show that this gives the transforming functiong(x) =− 1
λln(1−
x).
(b) On your TI calculators, extract 100 random samples of the
exponential distribution withλ=.5 (so μ= 2) via the com-
mand− 1. 5
ln(1−rand(100))→L 1.
This will place 100 samples into the list variableL 1.
(c) Draw a histogram of these 100 samples—does this look right?- (This is an extended exercise.) Continuing on the theme in Exer-
cise 7, we can similarly use the TI calculator to generate samples