:
x>0 a=const, a>0,
-: :
Y | a | |
P | P(x<a) | P(x a) |
X Y
M(x) M(y), M(y)= aP(X a)
aP(X a) M(x)
P(X a)
a=4 (.. a=const), a>0, M(x)=m
: P(X 4) m/4
7. , X Y , , (XY)=(X)(Y)
X Y , ( ).
X x1, x2, , Y - y1, y2, pij=P(X=xi, Y=yj). XY . M(XY)= X Y : P(X= xi, Y=yj)= P(X=xi) P(Y=yj). P(X=xi)=ri, P(Y=yj)=sj, pij=risj
, M(XY) = = . , : M(XY)=()() = M(X)M(Y)
8. , X Y , , (X + Y) = (X) +(Y).
:M(X+Y)= M(X)+M(Y). -. X Y (*)( 2 ):
X x1 x2 p p1 p2 | Y y1 y2 g g1 g2 |
X+Y. X Y; x1+y1, x1+y2, x2+y1, x2+y2. , ( , ), p11,p12,p21,p22. X+Y : M(X+Y) = (x1+y1)* *p11+(x1+y2)* p12+(x2+y1)* p21+(x2+y2)* p22, M(X+Y) = x1*(p11+p12)+ x2*(p21+p22)+ +y1*(p11+p21)+ y1*(p12+p22). , p11+p12=p1. , , X x1 ( p1), , , X+Y x1+y1 x1+y2 ( p11+p12), . , p11+p12=p1. p21+p22=p2, p11+p21=g1 p12+p22=g2. (*), M(X+Y)=(x1p1+x2p2)+(y1g1+y2g2), M(X+Y)= M(X)+M(Y).
9. , n . , ()=n, D()=n(1-).
n , , Ā q=1-p. . n . : =1+2++n. , (i)=, D(i)=np. . , :
|
|
q |
, ()=, 2 , D()=(2)-2()=-2=(1-)=q. , (i)=, D(i)=pq. ()=(1)+..+(n)=n, D()=D(1)++D(n)=npq=np(1-).
10. X , . , M (X) = D (X) = λ.
:
X | |||||
P | λ−e | λλ−e | λλ−e!22 | λλ−e!33 |
:
=
, λ, , X. , , , , ∞→n∞→nλ = np. np, , M(X) =. , , X λ, D(X) = npq 1 →q. , , - . . ,
M(X) = λ, D(X) = λ
11. , . , M (X) = 1/.
1- (), =()
n | |||||
pq | Pqn-1 |
12. , .
. : pxy = Kxy / . , = = . , . .
1. [-1;1],.. -1< <1.
As Ex , , , , As Ex .
As=μ3/σ3, σ=√D, μ3=M[(x-M(x)3]
Ex= μ4/σ4-3
fx=1/(b-a)=1, μ3= Sb a fx(t)tdt== Sb a tdt=t2/2 a =(b-a)2/2
D== Sb a fx(t)t2dt=(b-a)3/3
σ=√D=√(b-a)3/3
As=μ3/σ3=((b-a)2/2)/(√(b-a)3/3)
Ex= μ4/σ4-3=((b-a)5 /5)/((b-a)3/3)2 - 3
μ4= M[(x-M(x)4] fx(t)tdt= Sb a t4dt=(b-a)5 /5
f (x)? , X D(X) , M(X) .