Q.4.5 - A First Course in Probability

Question

Let N be a nonnegative integer-valued random variable. For nonnegative values aj, j Ú 1, show that 

$\sum _{j=1}^{\infty }\left(a_1+\dots +a_j\right)P\{N=j\}=\sum _{i=1}^{\infty }{a}_{i}P[N\ge i\}$

Then show that 

$E\left[N\right]=\sum _{i=1}^{\infty }P[N\ge i\}$

and

$E\left[N\right(N+1\left)\right]=2\sum _{i=1}^{\infty }iP\{N\ge i\}$

Step-by-Step Solution

Answer

In the given information the answers are

$\sum _{j=1}^{+\infty }\left(a_1+\dots +a_j\right)P(N=j)=\sum _{i=1}^{+\infty }{a}_{i}P(N\ge i)$

$E\left(N\right)=\sum _{i=1}^{+\infty }P(N\ge i)$

$E\left(N\right(N+1\left)\right)=2\sum _{i=1}^{+\infty }iP(N\ge i)$ proved

Step 1:Given Information (1)

N is a non negative integer valued random variable . 

Step 2:Calculation (1)

 $\sum _{j=1}^{+\infty }\left(a_1+\dots +a_j\right)P(N=j)$

$=\sum _{j=1}\left(a_{1}P(N=j)+\dots +a_{j}P(N=j)\right)$

$=a_{1}P(N\ge 1)+a_{2}P(N\ge 2)+a_{3}P(N\ge 3)+\dots$

$=\sum _{i=1}^{+\infty }{a}_{i}P(N\ge i)$ Hence it is proved.

        

Step 3:Given Information (2)

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

$E\left(N\right)=\sum _{x=1}^{+\infty }xP(N=x)$

Step 4 :Calculation (2)

$E\left(N\right)=\sum _{x=1}^{+\infty }xP(N=x)$ $=P(N=1)+2P(N=2)+3P(N=3)+4P(N=4)+\dots$

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

$=P(N\ge 1)+P(N\ge 2)+P(N\ge 3)+P(N\ge 4)+\dots$

$=\sum _{i=1}^{+\infty }P(N\ge i)$Hence it is proved

Step 5:Given Information (3)

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

$E\left(N\right(N+1\left)\right)=\sum _{x=1}^{+\infty }x(x+1)P(N=x)$

Step 6 :Calculation (3)

$E\left(N\right(N+1\left)\right)=\sum ^{+\infty }x(x+1)P(N=x)$ 

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

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

$=2P(N\ge 1)+4P(N\ge 2)+6P(N\ge 3)+8P(N\ge 4)+\dots$

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

$=2\sum _{i=1}^{+\infty }iP(N\ge i)$ Hence it is proved

Step 7:Final Answers

The final answers are$\sum _{j=1}^{+\infty }\left(a_1+\dots +a_j\right)P(N=j)=\sum _{i=1}^{+\infty }{a}_{i}P(N\ge i)$,

$E\left(N\right)=\sum _{i=1}^{+\infty }P(N\ge i)$,

$E\left(N\right(N+1\left)\right)=2\sum _{i=1}^{+\infty }iP(N\ge i)$ Are proved