5. Random vectors and Joint Probability Distributions
温暖的反义词 随机向量与联合概率分布
5.1 Concept of Joint Probability Distributions
(1) Discrete Variables Ca 离散型
Often, trials are conducted where two random variables are obrved simultaneously in order to determine not only their individual behavior but also the degree of relationship between them.
( X, Y)
For two discrete random variables X and Y, we write the probability that X will take the value x and Y will take the value y as P(X=x, Y=y). Conquently, P(X=x, Y=y) is the probability of the interction of the events X=x and Y=y.
(X=x, Y=y) ------ (X=x)∩(Y=y)
The distribution of probability is specified by listing the probabilities associated with all possible pairs of values x and y, either by formula or in a table. We refer to the function p(x, y)= P(X=x, Y=投影仪支架y) and the corresponding possible values (X, Y) as the j业主大会和业主委员会指导规则oint probability distribution (联合分布)of X and Y.
X YX | y1 | y2 | … | yj | … |
x1 | p11 | p12 | … | p1j | |
x2 | p21 | p22 | … | p2j | |
… | | | | | |
Xi | pi1 | pi2 | … | pij | … |
… | | | 退位减法怎么教 | | |
| | | | | |
They satisfy
,
where the sum is over all possible values of the variable.
Example 5.1.1 Calculating probabilities from a discrete joint probability distribution
Let X and Y have the joint probability distribution.
巨蟹金牛X Y | 0 1 |
0 1 2 | 0.1 0.2 0.垃圾怎么分类4 0.2 0.1 0 |
| |
(a) Find ;
(b) Find the probability distribution of the individual random variable X.
Solution
(a) The event is compod of the pairs of values (l,1), (2,0), and (2,l). Adding their corresponding probabilities
(b) Since the event X东北凉菜做法大全=0 is compod of the two pairs of values (0,0) and (0,1), we add their corresponding probabilities to obtain
.
Continuing, we obtain and
.
In summary, , and is the probability distribution of X.
Note that the probability distribution of appears in the lower margin of this enlarged table. The probability distribution of Y appears in the right-hand margin of the table. Conquently, the individual distributions are called marginal probability distributions.（边缘分布）
X Y | 0 1 | pX(x) |
0 1 2 | 0.1 0.2 0.4 0.2 0.1 0 | 0.3 0.6 0.1 |
pY(y) | 0.6西山公园 0.4 | 1.0 |
| | |
From the example, we e that for each fixed value of x, the marginal probability distribution is obtained as
,
where the sum is over all possible values of the cond variable. Continuing, we obtain
.
Example 3.5.3
Suppo the number of patent applications （专利申请）submitted by a company during a 1-year period is a random variable having the Poisson distribution with mean , ()and the various applications independently have probability of eventually being approved.
