5. Формула Бернуллі. Біноміальний закон розподілу.

Formula of Bernoulli. Binomial law of distribution.

During the tests event A can occur with probability P(A)=p and non-occur with probability P(A̅)=1-P(A)=q.

S uppose that n independent tests are made. It is necessary to find probability, that event A occurs m times in n tests. This probability is denoted P_n(m). So event A occur n times and non-occur n-m times. Also we denoted that the order of occurrence and non-occurrence is not important. Event A in n outcomes can occur m times in different order (or combinations) number of these outcomes is equal C_n^m.

L et event B is combination of event A and A̅.

B=A₁,A₂,A₃,…A_m, A̅_m+1, A̅_m+2,…A̅_n

m times n-m times

P(B)=P(A1)*(PA2)…P(Am)P(Am+1)P(A_m+2)…P(A_n)=p^m(1-p)^n-m=p^mq^n-m

So the formula of Bernoulli is: P_n(m)=C^m_np^mq^n-m

6. Найімовірніша частота та її імовірність

The most probable number of occurrence and its probability.

The most probable of occurrence of event A in n tests is denoted by m₀. M₀ is occurrence of event A in n independent tests, which possibility is P_n(m₀) at least not less than possibility of other event P_n(m) for any m.

Let’s find its possibility P_n(m₀)

M₀ can be evaluated by formula m₀≈n*p

M₀ can get 1 or 2 values as integer. P_n(m₀).

7. Формула Пуассона. Закон рідкісних подій

Poisson’s formula

Theorem: if possibility of occurrence of event A in each test aims to zero p->0 in unlimited increased number of tests (n->∞) and product n*p aims to constant number a, then possibility P_n(m) of occurrence of event A equal m times in n independent tests is equal:

Lim P_n(m)=a^me^-a

^n->∞ m!

Suppose, that p->0

n-> ∞

n*p->a

For n>=100, a=n*p=<10, than for finding P_n(m) we use Poisson’s formula

P_n(m)=a^me^-a

m!

This formula also is called as low of rare events.

9. Математичне сподівання дискретної випадкової величини та його властивості. Довести (на вибір) дві з них.

Mathematical expectation of DRV and its properties

Mathematical expectation of DRV X is the sum of product of all values of X their

_n

probabilities. It’s denoted by M(X) = ∑x_i*p_i

ⁱ⁼¹

Properties of M.E.

1. M(C)=C, c= const

x	C	…	C
p	p1	…	pn

Proof: Lets

M (C)=C*p₁+C*p₂+…+C*p_n = C(p₁+p₂+…p_n) = C

1

2. Constant multiplier can be taken out of the brackets: M(C*X)=C*M(X)

Proof

x	x1	x2	…	xn
p	p1	p2	…	pn

Cx	Cx1	Cx2	…	Cxn
p	p1	p2	…	pn

M (CX)=CX₁*p₁+CX₂*p₂+…+Cx_n*p_n=C(x₁p₁+x₂p₂+…+x_np_n)=C*M(X)

M(X)

3. M(X­+Y)=M(X)+M(Y); M(X-Y)=M(X)-M(Y)

4. For independent DRV M(X*Y)=M(X)*M(Y)

5. M(X-M(X))=0