Mathematical expectation is also called expectation, expectation value, etc. In probability theory and statistics, the expected value of discrete random variables is the sum of the probability of each possible result multiplied by its result in the experiment.
What does this mean? If we play a game, there are 52 cards, including 4 aces. We bet 1 yuan. If you win A, I will give you 10 yuan, otherwise your 1 yuan will be lost to me. In this game, the winning probability is113 (452)13 (452), and the result is 10 yuan; The failure probability is 1 2131213, and the result is a loss of1yuan. So what is your probability of winning, that is, what is your expectation? 2 13? 2 13。 In this way, after you play too much, you will find that everyone will lose money on average. 2 13 ? 2 13 yuan. Generally speaking, in a game, if X is a discrete random variable, its possible values are X 1, X2X 1, X2 ..., and its corresponding probabilities are P 1, P2P 1, P2 ..., and the sum of the probabilities is/kloc-.
For mathematical expectations, we should also clarify some knowledge points:
(1) Expected "Linear" attribute. For all discrete random variables X, Y and constants A and B, there are: e (ax+by) = AE (x)+Be (y) E (ax+by) = AE (x)+Be (y); Similarly, we also have E(XY)=E(X)+E(Y)E(XY)=E(X)+E(Y).
(2) Total probability formula? Suppose {bn ∣ n = 1, 2,3, ... ∣ n = 1, 2,3, ...} is a finite probability space or a countable infinite partition, and the set BnBn is a countable set, then for any event A, there are
p(a)=∑np(a∣bn)p(bn)p(a)=∑np(a∣bn)p(bn)
(3) Complete expectation formula? e(y)=e(e(y∣x))=∑ip(x=xi)e(y∣x=xi)
2. Variance: Variance is a measure of the oscillation degree near the expected μ=E(X)μ=E(X) (mean), which can be calculated by the following formula.
Var(X)=E(X？ μ)2
Var(X)=E(X？ μ)2
An equivalent formula is:
Var(X)=E(X2)？ E2(X)
Var(X)=E(X2)？ E2(X)
The nature of variance:
( 1)? Var (x) ≥ 0var (x) ≥ 0, and var (c) = 0var (c) = 0, indicating that the constant does not fluctuate.
(2)? Var(cX)= c2Var(X)Var(cX)= c2Var(X)？ This formula provides a way to improve oscillation, that is, to expand and contract the value of random variables.
(3)? Var(X+c)=Var(X)Var(X+c)=Var(X), moving the values of all the following variables will not change the degree of oscillation.
(4)? The variance of the sum of independent random variables is equal to the sum of variance (note: this property of mean does not require the independence of random variables)
Var(X+Y)=Var(X)+Var(Y)
Var(X+Y)=Var(X)+Var(Y)
Prove:
Var(X+Y)=E(X2+Y2+2XY)？ E2(X)？ E2(Y)？ 2E(X) east (y)
Var(X+Y)=E(X2+Y2+2XY)？ E2(X)？ E2(Y)？ 2E(X) east (y)
Because x, yx and y are independent of each other.
E(XY)=E(X)E(Y)
E(XY)=E(X)E(Y)
Substitute in the above formula and you will get it.
Var(X+Y)=Var(X)+Var(Y)
Var(X+Y)=Var(X)+Var(Y)
Judging from the process of proof, independent conditions are essential. Since variance is defined by expectation, all the properties of variance can be derived from expectation, which shows that the concept of expectation is more important than variance.