91Ó°ÊÓ

N is a non negative integer valued random variable .

$\underset{j=1}{\overset{+∞}{∑}}\left(a_1+…+a_j\right)P(N=j)$

$=\underset{j=1}{∑}\left(a_{1}P(N=j)+…+a_{j}P(N=j)\right)$

$=a_{1}P(Nâ‰\yen 1)+a_{2}P(Nâ‰\yen 2)+a_{3}P(Nâ‰\yen 3)+…$

$=\underset{i=1}{\overset{+∞}{∑}}{a}_{i}P(Nâ‰\yen i)$ Hence it is proved.

The expected value is the sum of he product of each possibility x with its probability P (x):

$E\left(N\right)=\underset{x=1}{\overset{+∞}{∑}}xP(N=x)$

$E\left(N\right)=\underset{x=1}{\overset{+∞}{∑}}xP(N=x)$$=P(N=1)+2P(N=2)+3P(N=3)+4P(N=4)+…$

$=\left(P\right(N=1)+P(N=2)+P(N=3)+P(N=4)+…)$

$=P(Nâ‰\yen 1)+P(Nâ‰\yen 2)+P(Nâ‰\yen 3)+P(Nâ‰\yen 4)+…$

$=\underset{i=1}{\overset{+∞}{∑}}P(Nâ‰\yen i)$Hence it is proved

The expected value is the sum of product of each possibility x with its probability P(x):

$E\left(N\right(N+1\left)\right)=\underset{x=1}{\overset{+∞}{∑}}x(x+1)P(N=x)$

$E\left(N\right(N+1\left)\right)=\stackrel{+∞}{∑}x(x+1)P(N=x)$

$=2P(N=1)+6P(N=2)+12P(N=3)+20P(N=4)+…$

$=\left(2P\right(N=1)+2P(N=2)+2P(N=3)+2P(N=4)+…)$

$=2P(Nâ‰\yen 1)+4P(Nâ‰\yen 2)+6P(Nâ‰\yen 3)+8P(Nâ‰\yen 4)+…$

$=2\left(P\right(Nâ‰\yen 1)+2P(Nâ‰\yen 2)+3P(Nâ‰\yen 3)+4P(Nâ‰\yen 4)+…)$

$=2\underset{i=1}{\overset{+∞}{∑}}iP(Nâ‰\yen i)$Hence it is proved

The final answers are$\underset{j=1}{\overset{+∞}{∑}}\left(a_1+…+a_j\right)P(N=j)=\underset{i=1}{\overset{+∞}{∑}}{a}_{i}P(Nâ‰\yen i)$,

$E\left(N\right)=\underset{i=1}{\overset{+∞}{∑}}P(Nâ‰\yen i)$,

$E\left(N\right(N+1\left)\right)=2\underset{i=1}{\overset{+∞}{∑}}iP(Nâ‰\yen i)$Are proved