Introduction to Probability and Statistics for Engineers and Scientists

160 Chapter 5: Special Random Variables

where the next-to-last equality used the fact thatX+Y is binomial with parameters
(n+m,p). Hence, we see that the conditional distribution ofXgiven the value ofX+Y
is hypergeometric.
It is worth noting that the preceding is quite intuitive. For suppose thatn+minde-
pendent trials, each of which has the same probability of being a success, are performed;
letXbe the number of successes in the firstntrials, and letYbe the number of successes
in the finalmtrials. Given a total ofksuccesses in then+mtrials, it is quite intuitive
that each subgroup ofktrials is equally likely to consist of those trials that resulted in
successes. That is, theksuccess trials are distributed as a random selection ofkof the
n+mtrials, and so the number that are from the firstntrials is hypergeometric. ■

5.4The Uniform Random Variable

A random variableXis said to be uniformly distributed over the interval[α,β]if its
probability density function is given by

f(x)=







1

β−α

ifα≤x≤β

0 otherwise

A graph of this function is given in Figure 5.4. Note that the foregoing meets the
requirements of being a probability density function since

1

β−α

∫β

α

dx= 1

The uniform distribution arises in practice when we suppose a certain random variable is
equally likely to be near any value in the interval[α,β].
The probability thatXlies in any subinterval of[α,β]is equal to the length of that
subinterval divided by the length of the interval[α,β]. This follows since when [a, b]

f(x)

a b

1

b – a

x

FIGURE 5.4 Graph of f(x)for a uniform[α,β].