MATH 210 – Lecture 1
Finite Mathematics
Semester I, 2007-2008
CONTINUOUS RANDOM VARIABLES
This handout augments the material in Section 4.3 of the text.
Often, a random variable Xhas a continuum of possible values, rather
than taking values from a finite set, and the probability that Xis equal to
any specific value is zero. In theory, Xmay take on any real number as a
value, although in practice, we often have some bounded range into which
we’re 100% sure that Xwill fall.
For example, if Xrepresents the weight of a random human in pounds,
we don’t have a finite list of possible values of X. It’s fairly safe to say
that Pr[0 ≤X≤10000] = 1. It’s also safe to say that the probability that
you weigh a specific value, say, exactly 165.2319493214 pounds, is essentially
zero; in symbols, Pr[X=a] = 0 for each a.
Let f(x) be a function such that f(x)≥0 for all xand the total area
between the xaxis and the graph of fis exactly 1. We say that a random
variable Xhas probability density function f(x) if Pr[X=a] is always 0,
and, when a < b, Pr[a≤X≤b] is the area bounded by the xaxis, the graph
of f, and the lines x=aand x=b.
123
−1
0.4
−2
In this picture, we define f(x) to be 12
625 (2 + x)(3 −x)(1 + x2) for −2≤x≤3
and 0 for other values. Then the area under the entire curve fis 1, and
Pr[1 ≤X≤2] is exactly 0.32096 (you can see by eye that the shaded area
is approximately 1/3 the total area). Unfortunately, to actually compute
these values, you have to be able to compute the area under a curve given a
formula for the curve. You learn that in calculus (take Math 211).
In this course, we’ll only study examples where you can compute the area
either by elementary geometry (see the example below), or by using the table
in Appendix A of the text for the standard normal curve (see Section 4.3 for
examples).
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