5. Random vectors and Joint Probability Distributions
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,七夕英语 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=美国经典战争大片y). Conquently, P(X=x, nipple是啥意思 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=y) and the corresponding possible values (X, Y) as the joint probability distribution (联合分布)of X and Y.
X 小学一年级语文课本YX | y1 | y2 | … | yj | … |
x1 | p11 | p12 | … | pressuresp1j | |
x2 | p21 | p22 | … | p2j | |
… | | | | | |
Xi | pi1 | stationery是什么意思pi2 | … | pij | … |
菲利普霍夫曼… | | | | | |
教育部不建议占用假期补课 | | | | | |
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.
X Y | 0 1 |
0 1 2 | 0.1 0.2 0.4 0.2 dcd 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=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）
| 外语人才网 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 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.
