358 CHAPTER 6 Inferential Statistics
circles intersect. (Hint: letX 1 andX 2 be independent instances of
randand note that it suffices to computeP(4(X 1 −X 2 )^2 + 1≤4).)- LetX be a random variable with density functionf(x), non-zero
on the intervala≤x≤b. Compute the density function ofcX+d,
wherecanddare constants witha >0, in terms off. - Define the random variableZ=rand, and so 0≤Z≤1.
(a) Determine the density function ofY = 10Z.
(b) Notice that the random variableY satisfies 1≤Y ≤10. Show
that the probability that a random sample ofY has first digit
1 (i.e., satisfies 1≤Y <2) is log 102 ≈30.1%. (This result is
a simplified version of the so-calledBenford’s Law.)
(c) Data arising from “natural” sources often satisfy the property
that their logarithms are roughly uniformly distributed. One
statement of Benford’s Law is that—contrary to intuition—
roughly 30% of the data will have first digit 1. We formal-
ize this as follows. Suppose that we a random variable 1 ≤
Y ≤ 10 n, where n is any positive integer, and assume that
Z = log 10 Y is uniformly distributed. Show that the proba-
bility that a random sample ofY has digit d, 1 ≤ d ≤ 0 is
log 10(
1 +1
d)6.2.3 The exponential distribution
Theexponential random variableis best thought of as a continuous
analog of the geometric random variable. This rather glib statement
requires a bit of explanation.
Recall that if X is a geometric random variable with probability
p, then P(X = k) is the probability that our process (or game) will
terminate afterkindependent trials. An immediate consequence of this
fact is that the conditional probabilities P(X =k+ 1|X ≥ k) = p,
and hence is independent of k. In other words, if we have managed
to survive k trials, then the probability of dying on thek+ 1-st trial
depends only on the parameterpand not onk. Similarly, we see that
P(X=k+τ|X≥k) will depend only onpand the integerτ, but not