Independent events

The standard definition says:

Here A ∩ B is the intersection of A and B, that is, it is the event that both events A and B occur.

More generally, any collection of events—possibly more than just two of them—are mutually independent if and only if for every finite subset A₁, ..., A_n of the collection we have

This is called the multiplication rule for independent events. Notice that independence requires this rule to hold for everysubset of the collection; see^[2] for a three-event example in which   and yet no two of the three events are pairwise independent.

If two events A and B are independent, then the conditional probability of A given B is the same as the unconditional (or marginal) probability of A, that is,

There are at least two reasons why this statement is not taken to be the definition of independence: (1) the two events A and B do not play symmetrical roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved.

The conditional probability of event A given B is given by

The statement above, when   is equivalent to

which is the standard definition given above.

Note that an event is independent of itself if and only if

That is, if its probability is one or zero. Thus if an event or itscomplement almost surely occurs, it is independent of itself. For example, if event A is choosing any number but 0.5 from auniform distribution on the unit interval, A is independent of itself, even though, tautologically, A fully determines A.

30. Mathematical expectation

Example: we have 3 red balls and 4 blue balls in a box. At randomly getting we choose 2 balls. Find the probability that 2 balls are red.

Sample Space	X
RR	2
RB	1
BR	1
BB	0

X-random variable

P(X)= 2* 2/7 + 1* 2/7 + 0*0= 6/7

Def.1. A random variable is a function what associated a real number with each element is a sample.

Def.2. The set of ordered pairs (x, f(x)) is a probability function or probability mass function or probability distribution of a discrete random variable, if for each possible outcomes x is equal

1. f(x) ≥0

2. ∑(x=0, to ∞ )=1

3. P(X=x)=f(x)

Def.3. Let X is a random variable with probability distribution f(x). The mean or expected value of x

µ=E(X)= ∑x f(x)

Def 4 Let x be a random variable with probability distribution f (x) and expected value µ the variance of x is equal

σ ² =E(x-µ)²= ∑ (x-µ)f(x)

def 5 Let x random variable with probability f(x) and expected value µ the variance of x= σ ²

σ ² =E(x-µ)²=∑(x-µ)² f(x)

def 6 let x and y random variable with joint probability distribution f(x,y) the covariance of x,y we denote

σ _x,y= E(x-µ_x)(y-µ_y)=∑(x-µ_x)(y-µ_y) f(x,y)

def 7 let x,y be a random variable with covariance σ _x,y and standart deviation σ_x and σ_y respectively. The correlation:

ƥ _xy=

Theorem 2 the variance of a random variable x is

σ²=E (x ²)-µ²