Chapter 5 Discrete Distributions

110 CHAPTER 5. DISCRETE DISTRIBUTIONS
> F = cumsum(f)
> F
[1] 0.125 0.500 0.875 1.000
As easy as this is, it is even easier to do with the distrEx package [74]. We define a
random variable X as an object, then compute things from the object such as mean, variance,
and standard deviation with the functions E, var, andsd:
> library(distrEx)
> X E(X); var(X); sd(X)
[1] 1.5
[1] 0.75
[1] 0.8660254
5.2 The Discrete Uniform Distribution
We have seen the basic building blocks of discrete distributions and we now study particular
models that statisticians often encounter in the field. Perhaps the most fundamental of all is the
discrete uniform distribution.
ArandomvariableX with the discrete uniform distribution on the integers 1, 2,...,m has
PMF
f X (x) = 1 , x = 1, 2,...,m. (5.2.1)
m
We write X ∼ disunif(m). A random experiment where this distribution occurs is the choice
of an integer at random between 1 and 100, inclusive. Let X be the number chosen. Then
X ∼ disunif(m = 100) and
IP(X = x) = 1
100 , x = 1,...,100.
We find a direct formula for the mean of X ∼ disunif(m):
m∑
m∑
µ = xf X (x) = x · 1
m = 1 m + 1
(1 + 2 + ···+ m) =
m 2 , (5.2.2)
x=1
x=1
where we have used the famous identity 1 + 2 + ···+ m = m(m + 1)/2. That is, if we repeatedly
choose integers at random from 1 to m then, on the average, we expect to get (m + 1)/2. To get
the variance we first calculate
and finally,
IE X 2 = 1 m
m∑
x=1
σ 2 = IE X 2 − (IE X) 2 =
x 2 = 1 m(m + 1)(2m + 3)
=
m 6
(m + 1)(2m + 1)
6
( m + 1
−
2
(m + 1)(2m + 1)
,
6
) 2
= ···= m2 − 1
12 . (5.2.3)
Example 5.4. Roll a die and let X be the upward face showing. Then m = 6, µ = 7/2 = 3.5,
and σ 2 = (6 2 − 1)/12 = 35/12.