# Linearity Property of Expectation

• Jan 31st 2007, 08:02 PM
led5v
Linearity Property of Expectation
Could some one please provide me with a detailed proof of the following linear properties of Expectation. I found a couple of informal proofs for these on the internet which dont outline the fact that how are the constants taken out of the Event of the probability and the infinite summation.

Property 1: E(X+c)=E(X)+E(c)

Property 2: E(aX) = aE(X)$where E(X) = \sum_n n Pr(X=n) Thanks. • Jan 31st 2007, 08:57 PM CaptainBlack Quote: Originally Posted by led5v Could some one please provide me with a detailed proof of the following linear properties of Expectation. I found a couple of informal proofs for these on the internet which dont outline the fact that how are the constants taken out of the Event of the probability and the infinite summation. Property 1: E(X+c)=E(X)+E(c) I assume that c is a constant, is that right? By the definition of expectation, and the linearity of summation: $E(X+c)=\sum_n (x_n+c)\, Pr(X=n)=$ $\sum_n [x_n\, Pr(X=x_n)+ c\, Pr(X=x_n)]$ ...... $ =\sum_n x_n\, Pr(X=x_n)+\sum_n c\, Pr(X=x_n)=E(X)+E(c) $ Quote: Property 2: E(aX) = aE(X)$
Similarly:

$E(aX)=\sum_n a\,x_n\, Pr(X=x_n) =a\sum_n x_n\, Pr(X=x_n)=aE(X)$

Quote:

where E(X) = \sum_n n Pr(X=n)
Thanks.
RonL
• Feb 1st 2007, 11:45 AM
led5v
Thanks RonL for your reply and outlining the proof steps.

I still have one confusion about the steps that you mentioned. My understanding is that according to the definition of the Expectation E(X + c) would be equal to

$\sum_n n * (Pr (X + c = n))$

So I didn't quite understand how did you get

\$sum_n (n + c) Pr (X=n)$

in the first step of your proof sequence.

Similarly, in the second property, I dont understand how did you get to the first step using the definition of expectation, i.e, from $\sum_n n * (Pr (aX = n))$ to
$\sum_n a * n * (Pr (X = n))$
• Feb 1st 2007, 11:54 AM
CaptainBlack
Quote:

Originally Posted by led5v
Thanks RonL for your reply and outlining the proof steps.

I still have one confusion about the steps that you mentioned. My understanding is that according to the definition of the Expectation E(X + c) would be equal to

$\sum_n n * (Pr (X + c = n))$

So I didn't quite understand how did you get

$\sum_n (n + c) Pr (X=n)$

in the first step of your proof sequence.

Similarly, in the second property, I dont understand how did you get to the first step using the definition of expectation, i.e, from $\sum_n n * (Pr (aX = n))$ to
$\sum_n a * n * (Pr (X = n))$

In my book (metaphorical) after conversion to your notation:

$E(f(X))=\sum_n f(n)\, Pr(X=n)$,

and in your first example $f(n)=n+c$

In fact in your definition:

$E(X+c)=\sum_n n\,Pr (X + c = n)$

Don't you have a problem in that $Pr(X+c=n)$ is not defined for non-integer $c$ (since X is a discrete random variable taking integer values)? While $E(X+c)$ should be defined for any real or complex $c$

RonL
• Feb 1st 2007, 12:08 PM
led5v
Now, I am really confused :confused: ....It seems that all my understanding about the expectation that I developed in the past few days is wrong :( ...

In the definition that I gave, c, is also an integer and not a real number. But still I think that there is some problem with my definition now.

RonL: Could you please give me a formal definition of expectation and shed a little bit of light about all the parameters that are present there. I would greatly appreciate that. and thanks for all your help.
• Feb 1st 2007, 12:12 PM
CaptainBlack
Quote:

Originally Posted by led5v
Now, I am really confused :confused: ....It seems that all my understanding about the expectation that I developed in the past few days is wrong :( ...

In the definition that I gave, c, is also an integer and not a real number. But still I think that there is some problem with my definition now.

RonL: Could you please give me a formal definition of expectation and shed a little bit of light about all the parameters that are present there. I would greatly appreciate that. and thanks for all your help.

Best look at the Mathworld page on expectation. I would normaly suggest
the Wikipedia page, but in this case I think it is probably not suitable.

RonL