Download The probability density function of a continuous random

The probability density function (PDF) of a continuous random variable
• A random variable X is continuous if P (X = x) = 0 for every x.
• PDF (機率密度函數) of a continuous random variable. For a continuous
random variable X, if there exists a non-negative function f such that
Z b
P (a < X < b) =
f (x)dx for all a < b,
(1)
a
then f is called the probability density function (PDF) of X.
– (1) implies that the distribution of X can also be described using its
PDF.
Rb
– The notation a f (x)dx is called the integral of f from a to b (f 從
a 到 b 的積分), which is the area of region D in the following graph.
Example 1. Suppose that X has PDF


 0

2x/3
f (x) =
(3 − x)/3



0
The graph of f is given below.
1
f , where
if
if
if
if
x < 0;
0 ≤ x ≤ 1;
1 ≤ x ≤ 3;
x > 3.
Find P (1 < X < 3).
Sol. The area under the PDF curve when 0 < x < 3 is (3 − 1) × (2/3)/2 =
2/3, so P (1 < X < 3) = 2/3.
• Suppose that X is a continuous random variable, then the PMF of X is
not useful for characterizing the distribution of X.
• Suppose that X is a random variable.
– If X is discrete, then the distribution of X can be characterized using
its PMF.
– If X is continuous with PDF f , then the distribution of X can be
characterized using f .
• If X is continuous with PDF f , then the mean and variance of X can be
calculated using
Z ∞
E(X) =
xf (x)dx
−∞
and
Z
∞
(x − µ)2 f (x)dx,
V ar(X) =
−∞
where µ = E(X). Computing E(X) and V ar(X) using the above formulas
is beyond the scope of this course.
2