;④P{x=a}=0.
二、常见的三种连续型随机变量的概率分布
常用的三种连续型随机变量的概率分布是均匀分布、指数分布和正态分布.
(1)均匀分布
若连续型随机变量X的概率密度为
MATLAB提供的有关均匀分布的函数如下:
unifpdf(X,A,B)均匀分布的密度函数
unifcdf(X,A,B)均匀分布的累积分布函数
unifinv(P,A,B)均匀分布的逆累积分布函数
unirnd(A,B,m,n)均匀分布的随机数发生器
unifstat(A,B)均匀分布的数学期望与方差
其中X为随机变量,P为概率值,A,B为均匀分布参数,m和n为生成随机数矩阵的行数和列数.
(2)指数分布
如果随机变量X的概率密度为
其中
为常数,则称X服从参数为
的指数分布,记作X~e(
).
MATLAB提供的有关指数分布的函数如下:
exppdf(X,L)指数分布的密度函数
expcdf(X,L)指数分布的累积分布函数
expinv(P,L)指数分布的逆累积分布函数
exprnd(X,L,m,n)产生服从指数分布的随机数
expstat(L)求指数分布的数学期望与方差
其中X为随机变量,L为参数
,P为显著概率,m和n为随机数矩阵的行数和列数.
绘制指数分布密度函数和累积分布函数图形的程序如下
x=-0.1:
0.001:
0.4;
y=exppdf(x,0.05);z=expcdf(x,0.05);
subplot(1,2,1);plot(x,y,'k');
axis([-0.1,0.4,-0.1,21]);
subplot(1,2,2);plot(x,z,'k');
-0.1
0
0.1
0.2
0.3
0
5
10
15
20
-0.1
0
0.1
0.2
0.3
0
0.5
1
axis([-0.1,0.4,-0.1,1.1]);
指数分布的密度函数及累积分布函数图
(3)标准正态分布
如果随机变量X的概率密度为:
其中
和
均为常数,且
>0,则称X服从参数为
和
2的正态分布,记作X~N(
2).当
=0,
=1时,称X服从标准正态分布,记作X~N(0,1).
MATLAB提供的有关正态分布的函数如下:
normpdf(X,M,C)正态分布的密度函数
normcdf(X,M,C)正态分布的累积分布函数
norminv(P,M,C)正态分布的逆累积分布函数
normrnd(M,C,m,n)产生服从正态分布的随机数
normstat(M,C)求正态分布的数学期望和方差
其中X为随机变量,M为正态分布参数
,C为参数
,P为显著概率,m和n为随机矩阵的行数和列数.
绘制标准正态分布的密度函数及累积分布函数图和一般正态分布的密度函数及累积分布函数图的程序如下:
-4
-2
0
2
4
0
0.2
0.4
-4
-2
0
2
4
0
0.5
1
0
5
10
15
0
0.2
0.4
0.6
0
5
10
15
0
0.5
1
x=-4:
0.01:
4;
y=normpdf(x,0,1);z=normcdf(x,0,1);
subplot(2,2,1);plot(x,y,'k');
axis([-4,4,-0.1,0.5]);
subplot(2,2,2);plot(x,z,'k');
axis([-4,4,-0.1,1.1]);
x=-4:
0.01:
16;
y1=normpdf(x,6,1);z1=normcdf(x,6,1);
y2=normpdf(x,6,4);z2=normcdf(x,6,4);
y3=normpdf(x,6,0.6);z3=normcdf(x,6,0.6);
subplot(2,2,3);plot(x,y1,'k',x,y2,'k',x,y3,'k');
axis([-4,16,-0.1,0.8]);
subplot(2,2,4);plot(x,z1,'k',x,z2,'k',x,z3,'k');
axis([-4,16,-0.1,1.1]);
三、求解方法
(1)通用函数介绍.
Pdf计算已选函数的概率密度函数,调用格式为:
Y=Pdf(name,X,A)
Y=Pdf(name,X,A,B)
Y=Pdf(name,X,A,B,C)
Name为上表中取stat后的字符,如beta、bino、chiz、exp等。
(2)利用专用函数.
Betapdf(X1,A1,B)
Binopaf(X,N,P)
四、例题
绘制卡方分布密度函数在n分别等于1,5,15时的值
>>x=0:
0.2:
30
y1=chi2pdf(x,1)
plot(x,y1,'+')
holdon
y2=chi2pdf(x,5)
plot(x,y2,'+')
y2=chi2pdf(x,15)
plot(x,y2,'o')
axis([0,30,0,0.2])
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