Measure, Integration & Real Analysis, 2021a

Chapter 12 Probability Measures 395
The equation ˜X(s) = ∫ s
−∞ h dλ holds by the definitions of ˜X and P. Thus h
Proof
is the density function of X.
Our definition of P to equal h dλ implies that ∫ ∞
−∞ f dP = ∫ ∞
−∞
fhdλ for all
f ∈L 1 (P) [see Exercise 5 in Section 9A]. Thus the formula for the mean EX
follows immediately from the definition of EX, and the formula for the variance
σ 2 (X) follows from 12.20.
The following example illustrates the result above with a few especially useful
choices of the density function h.
12.34 Example density functions
• Suppose h = 1 [0,1] . This density function h is called the uniform density on
[0, 1]. In this case, P(B) =λ(B ∩ [0, 1]) for each Borel set B ⊂ R. For the
corresponding random variable X(x) =x for x ∈ R, the distribution function
˜X is given by the formula
⎧
⎪⎨ 0 if s ≤ 0,
˜X(s) = s if 0 < s < 1,
⎪⎩
1 if s ≥ 1.
The formulas in 12.33 show that EX = 1 2
• Suppose α > 0 and
h(x) =
and σ(X) =
1
2 √ 3 .
{
0 if x < 0,
αe −αx if x ≥ 0.
This density function h is called the exponential density on [0, ∞). For the
corresponding random variable X(x) =x for x ∈ R, the distribution function
˜X is given by the formula
{
0 if s < 0,
˜X(s) =
1 − e −αs if s ≥ 0.
The formulas in 12.33 show that EX = 1 α and σ(X) = 1 α .
• Suppose
h(x) = 1 √
2π
e −x2 /2
for x ∈ R. This density function is called the standard normal density. For
the corresponding random variable X(x) =x for x ∈ R, wehave ˜X(0) = 1 2 .
For general s ∈ R, no formula exists for ˜X(s) in terms of elementary functions.
However, the formulas in 12.33 show that EX = 0 and (with the help of some
calculus) σ(X) =1.
Measure, Integration & Real Analysis, by Sheldon Axler