BV1ZX4y1K7M2

(资料图)

E(x)

=

Σn(1-p)^(n-1)p

=

pΣn(1-p)^(n-1)

=

p

1·(1-p)^0+2·(1-p)^1+3·(1-p)^2

+

...

+

n(1-p)^(n-1)

设

Sn

=

1·(1-p)^0+2·(1-p)^1+3·(1-p)^2

+

...

+

n(1-p)^(n-1)

有

(1-p)Sn

=

1·(1-p)^1+2·(1-p)^2+3·(1-p)^3

+

...

+

n(1-p)^n

即

-pSn

=

n(1-p)^n-1

-

(1-p)^1+(1-p)^2+(1-p)^3+...+(1-p)^(n-1)

=

n(1-p)^n-1

-

(1-p)(1-(1-p)^(n-1))/p

即

Sn

=

(1-(1+n)(1-p)^n)/p²

即

E(x)

=

(1-(1+n)(1-p)^n)/p

即

lim(n→+∞)E(x)

=

(1-lim(n→+∞)(1+n)(1-p)^n)/p

=

(1-lim(n→+∞)(1+n)/(1/(1-p)^n))/p

=

(1-lim(n→+∞)1/(-ln(1-p)/(1-p)^n))/p

=

(1-lim(n→+∞)(1-p)^n/(-ln(1-p)))/p

=

1/p

D(x)

=

E(x²)-E²(x)

=

pΣn²(1-p)^(n-1)-1/p²

=

p

1²·(1-p)^0+2²·(1-p)^1+3²·(1-p)^2

+

...

+

n²(1-p)^(n-1)

-

1/p²

设

Sn

=

1²·(1-p)^0+2²·(1-p)^1+3²·(1-p)^2

+

...

+

n²(1-p)^(n-1)

有

(1-p)Sn

=

1²·(1-p)^1+2²·(1-p)^2+3²·(1-p)^3

+

...

+

n²(1-p)^n

即

-pSn

=

n²(1-p)^n-1

-

3(1-p)^1+5(1-p)^2+7(1-p)^3

+

...

+

(2n-1)(1-p)^(n-1)

设

Tn

=

3(1-p)^1+5(1-p)^2+7(1-p)^3

+

...

+

(2n-1)(1-p)^(n-1)

有

(1-p)Tn

=

3(1-p)^2+5(1-p)^3+7(1-p)^4

+

...

+

(2n-1)(1-p)^n

即

-pTn

=

(2n-1)(1-p)^n-3(1-p)

-

2

(1-p)^2+(1-p)^3+...+(1-p)^(n-1)

=

(2n-1)(1-p)^n-3(1-p)

-

2

(1-p)²(1-(1-p)^(n-2))/p

=

((2n-1)p(1-p)^n-3(1-p)p)/p

-

(2(1-p)²-2(1-p)^n)/p

=

(2n-1+2/p)(1-p)^n+p+1-2/p

即

Tn

=

((1-2n)/p-2/p²)(1-p)^n-1-1/p+2/p²

即

-pSn

=

n²(1-p)^n-1

-

((1-2n)/p-2/p²)(1-p)^n-1-1/p+2/p²

=

(n²-(1-2n)/p+2/p²)(1-p)^n+1/p-2/p²

即

Sn

=

(-n²/p+(1-2n)/p²-2/p³)(1-p)^n-1/p²+2/p³

即

D(x)

=

(-n²+(1-2n)/p-2/p²)(1-p)^n-1/p+1/p²

即

lim(n→+∞)D(x)

=

lim(n→+∞)

(-n²-2n/p+(p-2)/p²)(1-p)^n+(1-p)/p²

=

lim(n→+∞)

(-n²-2n/p+(p-2)/p²)/(1/(1-p)^n)

+

(1-p)/p²

=

lim(n→+∞)

(-2n-2/p)/(-ln(1-p)/(1-p)^n)

+

(1-p)/p²

=

lim(n→+∞)

-2/(ln²(1-p)/(1-p)^n)

+

(1-p)/p²

=

(1-p)/p²

得证