Advanced High-School Mathematics

SECTION 6.2 Continuous Random Variables 349

the meaning of uniformity! What uniformity means is that for any two
numbersx 1 andx 2 and any small numbersatisfyingx 1 ±, x 2 ±∈
[0,1]

P(x 1 −≤rand≤x 1 +) =P(x 2 −≤rand≤x 2 +).

A much simpler description of the above is through the so-called
density functionfor the random variablerand. This has the graph
given below:

6

1

1

t

y=f(t)

The way to interpret this—and any other density curvey=f(x)—
is that the probability of finding a value of the corresponding random
variableXbetween the values aandbis simply thearea under the
density curvefromato b:

P(a≤X≤b) =

∫b

a f(x)dx.

For the uniform distribution this means simply that, for example,
P(rand≤ 2 /3) = 2/3, thatP(rand> .25) =.75,P(. 05 ≤rand< .6) =
.55, and so on.

Let’s consider another continuous random variable, defined in terms
of its density function. Namely, letX be the random variable whose
density functiony=fx(t) is as given below: