概率论与数理统计英文

46 / 17

4. Continuous Random Variable 连续型随机变量

Continuous random variables appear when we deal will quantities that are measured on a

continuous scale. For instanee, when we measure the speed of a car, the amount of alcohol in a pers

on's blood, the ten sile stre ngth of new alloy.

We shall lear n how to determ ine and work with probabilities relat ing to continu ous ran dom

variables in this chapter. We shall introduce to the concept of the probability density function.

4.1 Continuous Random Variable

1. Definition

Definition 4.1.1 A function f(x) defined on (-〜：)is called a probability density

function (概率密度函数)if:

(i) f (x) _0 for any x R;

oO

(ii) f(x) is in tergrable (可积的)on (-〜::)a nd f (x)dx = 1.

-nO

Definition 4.1.2

Let f(x) be a probability density function. If X is a random variable having distribution function

x

F(x)二 P(X 乞 x)二 f(t)dt, (4.1.1)

_oQ

then X is called a continuous random variable having density function f(x). In this case,

X2

P(m :: X :: x2) = f (t)dt. (4.1.2)

Xi

2. 几何意义

（）

x

F(x)二 P(X ^x)二 P((X,Y)|X zx, 0 乞丫 乞 f (X)) = f (t)dt

-oO47 / 17 x2

P(x, ::： X :：: x2) = f(t)dt

x1

3. Note

In most applications, f(x) is either

continuous or piecewise continuous having at

most fin itely many disc on ti nuities.

Note 1 For a random variable X, we have a

distribution function. If X is discrete, it has a probability distribution . If X is continuous, it has a

probability density function.

Note 2 Let X be a continuous random variable, then for any real number x,

P(X =x) =0.

0 < P(X =x) * f (x)dx

P(a 乞 X 乞 b)二 P(a 乞 X ::: b)二 P(a ：： X

4. Example

Example 4.1.2

Find k so that the following can serve as the probability density of a continuous random variable:

k

f(xr FW)

Solution To satisfy the conditions (4.1.1), k must be nonnegative, and to satisfy the condition

(4.1.2) we must have

oa

f (x)dx =

JOO

,1 so that k .

(Cauchy distribution 柯西分布)

Example 4.1.3 Calculating probabilities from the probability density function

If a ran dom variable has the probability den sity

Find the probability from that it will take on valueo 乞 p（x 二 X）空 lim x * ; f (x)dx = 0 ■ x

"x for x 0 f(x)= 3e [0 for x 乞 0 48 / 17

(a) betwee n 0 and 2;

(b) greater tha n 1.

Solution Evaluating the necessary integrals, we get

2

(a) P(0 辽 x 乞 2) = j3e'xdx =1 - e" =0.9975

0

oO

(b) P(x〉1) = 3e'Xdx = e" =0.0498

1

Example 4.1.4

Determ ining the distributi on function of X, it is known

工 3e'x for x 0 f(x)= 0 for x <0

Solution Performing the necessary integrations, we get

(0 for x _ 0 lx

"x) 3e"dt = 1—e"x for x 0

0

P(x _1) = F(1) =1 -e‘ =0.9502 □

5. mean

If the in tegral (4.1.3) does not con verges absolutely(绝对收敛 ),we say the mean of X does not

exist.

Definition 4.1.2 Let X be a continuous random variable having probability density function

f(x). Then the mean (or expectation) of X is defined by

—E(X)二.xf (x)dx , (4.1.3)

-oO

The mean of continuous random variable has the similar properties as discrete random variable.

If g(X) is an integrable function of a continuous random variable X, having density function f(x),

mean of g(X) is

oo

E(g(x))「g(x)f(x)dx

-JOO

provided the in tegral con verges absolutely.

Example 4.6.4

Let X be a random variable having Cauchy distribution, the probability density function is give n by

(a) Find E(X);二(1 x2) -：：::x 49 / 17

gw」ox：1

0, elsewhere

(b) Let

Find E(g(X)).

00 |x|

Solution (a) Since the integral ----------------- 2 dx diverges(发散), E(X) does not exist.

丿(1 + x )

1

(b) E(g(X))「g(x)f (x)dx =

_：: 0 x ln2 -------- 2 dx-——

二(1 x ) 2

二 6. variance

Similarly, the varianee and standard deviation of a continuous random variable X is defined by

二2 =D(X) =E((X 7)2), (4.1.4)

Where J = E(X) is the mean of X,二 is referred to as the standard deviation.

We easily get

□o

二2 二D(X) = x2f(x)dx-」2. (4.1.5)

Example 4.1.5

Determining the mean and varianee using the probability density funetion

3e"x for x 0

With referenee to the example 4.1.3: f (x)- < 0 for x 兰 0

find the mean and varianee of the given probability density.

Solution Performing the necessary integrations, we get

oO oo . 1 亠=xf(x)dx 二 x 3e'xdx =_ 0 3

and

c2 ：： 1 1

=J (x_ A)2 f(x)dx = J(x__)3e：Xdx =_

0 3 9

均匀分布 4.2 Uniform Distribution

The uniform distributi on, with the parameters a and b, has probability den sity fun eti on

一 for a x : b, f(x)pb-a

0 elsewhere,

whose graph is show n in Figure 4.2.1.

f(x)

1

b

—a