概率与统计课件第三章

=
∞ −∞
������
������, ������
������������
Examples
• Consider a circle of radius ������, and suppose that a point within the circle is randomly chosen in such a manner that all regions within the circle of
equal area are equally likely to contain the point. If we let the center of the circle denote the origin and define ������ and ������ to be the coordinates of the point chosen, then, since (������, ������) is equally likely to be near each point in the circle, it follows that the joint density function of ������ and ������ is given by
有房
无贷
10 %
下班按时 回家
60 %
好厨艺
20 %
• Joint Distribution Function（联合分布函数）
• Independent Random Variables（独立随机变量）
• Sums of Independent Random Variables(独立变量之和）
• Identically Distributed Uniform Random Variables（同分布均匀随机变量） • Gamma Random Variables（伽马随机变量） • Normal Random Variables（正态随机变量） • Poisson and Binomial Random Variables（泊松，二项随机变量） • Geometric Random Variables（几何随机变量）
Joint probability density function (联合密度函数)
• ������ and ������ are jointly continuous (联合连续) if there exists a function ������ ������, ������ , define for all ������ and ������ ,having the property that, for every set ������ of pairs of real numbers (that is, ������ is a set in the two-dimensional plane)
b
lim PX a,Y b b
lim F (a,b) F (a, ) b
FY (b) PY b
lim F (a,b) a
F (,b)
Properties
PX a,Y b 1 P(X a,Y bc ) 1 P(X ac Y bc ) 1 P(X a Y b) 1 P(X a) P(Y b) PX a,Y b
joint p.m.f. of ������ and ������
joint p.m.f. of ������ and ������
X Y
1
1
2
3
4
2
3
4
5
0 1/12 1/12 1/12
3
4
5
6
2 1/12 0 1/12 1/12
4
5
6
7
3 1/12 1/12 0 1/12
5
6
7
8
4 1/12 1/12 1/12 0
An extension
• Joint probability distributions for ������ random variable
• for any ������ sets of real numbers ������1, ������2, . . . , ������������
Independent random variables
• In terms of the joint distribution function ������ of ������ and ������, for all ������ and ������ F (a, b) FX (a)FY (b)
We want the p.m.f. of ������ + ������
1/12
0
1/12
In general P(X + Y = z) = ∑(x, y): x + y = z P(X = x, Y = y)
To calculate ������(������ + ������ = ������) we need to know f(x, y) = P(X = x, Y = y)
������ ������, ������ = ������ ������ ≤ ������, ������ ≤ ������ , −∞ < ������, ������ < ∞
• Marginal distributions (边缘分布)
FX (a) PX a PX a,Y P(limX a,Y b)
Joint probability density function
（联合密度函数）of ������ and ������
P(X ,Y ) C f (x, y)dxdy ( x, y )C
PX A,Y B f (x, y)dxdy
AB
2 f (a,b) F(a,b)
•（X,Y）
（0,0） R
Exercise
The joint density function of X and ������ is given by
f (x, y) e(xy) ,0x,0 y 0,otherwise
Find the density function of the random variable ������/������
1 FX (a) FY (b) F (a,b)
Pa1 X a2 ,b1 Y b2
F (a2 , b2 ) F (a1, b1) F (a1, b 2 ) F (a2, b1)
Joint probability mass function of ������ and ������ :
������ = pairs of cards, equally likely outcomes
������ = face value on first card
������ = face value on second card

P(X + Y = 4) = 1/6
= P(X = 1, Y = 3) + P(X = 2, Y = 2) + P(X = 3, Y = 1)
• Random variables ������ and ������ are said to be independent for any two sets of real numbers ������ and ������
PX A,Y B PX A PY B
• ������ and ������ are independent if, for all ������ and ������, the events ������������ = {������ ∈ ������} and ������������ = {������ ∈ ������} are independent
A
Probability density function of ������: ������������ ������
=
∞ −∞
������
������, ������
������������
Probability density function of ������: ������������ ������
f (x, y) c,ifx2 y2 R2 0,ifx2 y2 R2
(a) Determine ������ (b) Fine the marginal density functions of ������ and ������. (c) Compute the probability that ������, the distance from the origin of the point selected, is less than or equal to ������. (d) Find ������ [������].
ab
Marginal probability
• Probability density function of ������ from joint density function
PX A PX A,Y (, )

f (x, y)dydx
A
fX (x)dx
2
0
3 1/6
4 1/6
5 1/3
6 1/6
7 1/6
8
0
p.m.f. of X + Y
Joint distribution functions (联合分布函数)
• For any two random variables ������ and ������, the joint cumulative probability distribution function of ������ and ������ by