Sum of the Expected Values of Two Discrete Random Variables

  • #1
TheBigDig
65
2
Homework Statement:
Prove that E[aX+bY] = aE[X]+bE[Y]
Relevant Equations:
[tex]E[X] = \sum_{x=0}^{\infty} x p(x)[/tex]
Apologies if this isn't the right forum for this. In my stats homework we have to prove that the expected value of aX and bY is aE[X]+bE[Y] where X and Y are random variables and a and b are constants. I have come across this proof but I'm a little rusty with summations. How is the jump from the second line to the third line made?
1602533296609.png
 

Answers and Replies

  • #2
FactChecker
Science Advisor
Gold Member
7,450
3,224
The probability, ##P_X(x)##, for X taking a fixed value, x, is the sum over all values of Y of ##P_{XY}(x,y)##. So that is substituted in the third line. Likewise, the sum over all values of X of ##P_{XY}(x,y)## is replaced by ##P_Y(y)##.
 
  • #3
It's because$$\sum_y x P_{XY}(x,y) = x \sum_y P_{XY}(x,y) = xP_X(x)$$where the second equality follows from the law of total probability.

N.B. I think you're missing a ##\sum## before the ##y P_Y(y)##
 
  • #4
Stephen Tashi
Science Advisor
7,775
1,537
How is the jump from the second line to the third line made?

Have you studied joint distributions and their associated "marginal distributions"?

As an example, suppose ##P_{X,Y}## is given by
##P_{X,Y}(1,1) = 0.4##
##P_{X,Y} (1,2) = 0.2##
##P_{X,Y} (2,1) = 0.3##
##P_{X,Y}(2,3) = 0.1##

The associated marginal distribution for ##X## is:
##P_X(1) = 0.6 = P_{X,Y}(1,1) + P_{X,Y}(1,2)##
##P_X(2) = 0.4##

The term ##\sum_x \sum_y x P_{X,Y}(x,y)## denotes:

##( (1) ( P_{X,Y}(1,1) + (1)P_{X,Y}(1,2) ) + ( (2) P_{X,Y}(2,1) + (2)P_{X,Y}(2,2))##
## = (1) P_X(1) + (2)P_X(x)##

In the next line, the term ##\sum_x x P_X(x)## also denotes
## (1) P_X(1) + (2)P_X(2) ##
 
  • #5
TheBigDig
65
2
Have you studied joint distributions and their associated "marginal distributions"?

As an example, suppose ##P_{X,Y}## is given by
##P_{X,Y}(1,1) = 0.4##
##P_{X,Y} (1,2) = 0.2##
##P_{X,Y} (2,1) = 0.3##
##P_{X,Y}(2,3) = 0.1##

The associated marginal distribution for ##X## is:
##P_X(1) = 0.6 = P_{X,Y}(1,1) + P_{X,Y}(1,2)##
##P_X(2) = 0.4##

The term ##\sum_x \sum_y x P_{X,Y}(x,y)## denotes:

##( (1) ( P_{X,Y}(1,1) + (1)P_{X,Y}(1,2) ) + ( (2) P_{X,Y}(2,1) + (2)P_{X,Y}(2,2))##
## = (1) P_X(1) + (2)P_X(x)##

In the next line, the term ##\sum_x x P_X(x)## also denotes
## (1) P_X(1) + (2)P_X(2) ##

Okay yes, this definitely seems like something I need to read up on. Our instructor is a little handwavy at the moment saying we'll come across these concepts later but I'm one of those people who needs to understand each element.

It's because$$\sum_y x P_{XY}(x,y) = x \sum_y P_{XY}(x,y) = xP_X(x)$$where the second equality follows from the law of total probability.

N.B. I think you're missing a ##\sum## before the ##y P_Y(y)##

Thank you as well. Yes I think it is missing that. I found it online and just copied and pasted the image.
 

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