E[ ( aX + bY )^2 ] = E[ (aX)^2 + (bY)^2 ] ??

[M Willis]

Clearly, it is not true that

E[ ( aX + bY )^2 ] = E[ (aX)^2 + (bY)^2 ]

where X and Y are independent random variables and a and b are constants. However, in the solution to question 8 of the chapter 16 practice questions, it states that for a 10-year endowment policy with

• benefit 100,000 payable immediately on death; or
• benefit of 50,000 payable on survival of the 10-year term
• constant force of mortality of 0.03

the expected value of the present value squared is given by


Why is this?


P.S. Can I use LaTeX on these forums?

P.P.S what is the deal with only being able to attach 2 images to my post? If LaTeX is unsupported, then this is pretty unhelpful.

 

[Muppet]

https://acted.co.uk/forums/index.php?threads/writing-equations-in-your-posts.8070/

note the 'or' in the question.

 

[M Willis]

note the 'or' in the question.

Are you saying that if X and Y are mutually exclusive events then

E[ 2abXY ] = 0

 

[Muppet]

If X<>0 then Y = 0 and vice versa and so XY=0 at each point in time.
However, I wouldn't think of this as two separate random variables. It can be just one with amounts paid at various times in the future with different probabilities. You are just summing each outcome squared multiplied by the probability it will occur.