348 CHAPTER 6 Inferential Statistics
(b) Show that the maximum probability for a leader to be chosen
in a given round occurs whenp= 1/n.
(c) Show that ifn >>0, and ifp= 1/n, then the probability that
a leader is chosen in a given round is≈ 1 /e. (See Equation 6.7,
page 338.)- Suppose that in a certain location, the average annual rainfall is
regarded as a continuous random variable, and that the rainfalls
from year to year are independent of each other. Prove that inn
years the expected number of record rainfall years is given by the
harmonic series 1 +
1
2
+
1
3
+···+
1
n6.2 Continuous Random Variables
Your TI graphing calculator has a random-number generator. It’s called
rand; find it! Invoking this produces a random number. Invoking this
again produces another. And so on. What’s important here is that
rand represents a continuous (or nearly so!) random variable.
Let’s look a bit closer at the output of rand. Note first that the
random numbers generated are real numbers between 0 and 1. Next,
note that the random numbers are independent, meaning that the
value of one occurence of rand has no influence on any other occurence
ofrand.^12
A somewhat more subtle observation is that rand is a uniformly
distributedrandom variable. What does this mean? Does it mean
that, for example
P(rand=.0214) = P(rand= 1/π)?You will probably convince yourselves that this is not the meaning, as
it is almost surely true that both sides of the above are 0, regardless of
(^12) Of course, this isn’t the technical definition of “independence.” A slighly more formal definition
of independence of random variablesXandYis that
P(a≤X≤bandc≤Y≤d) =P(a≤X≤b)P(c≤Y≤d).