Year 12s: what will be the first way you research your future options? >>

Conbination of Random Variables - S3

Hi, I know this has probably been asked a lot, but I don't really understand it.
When combining random variables there are two cases, and I don't really understand when to use one case and when to use another, I don't think the S3 book explains it very well.
(i) E(aX +bY), Var(aX + bY) = aE(X) + bE(X), (a^2)Var(X) + (a^2)Var(Y)
(ii) But for linear combinations where X~N(mu1, sigmasquared1), Y~N(mu2, sigmasquared2),
aX +bY ~N(a*mu1 + b*mu2, a*sigmasquared1 + b*sigmasquared2)

Any help would really be appreciated,
thanks
Charlie

Reply 1

Contrad!ction.

Original post by charlie2434

I'm trying to ignore the formulae you've written because reading non-LaTeX stuff hurts my eyes :tongue:

I think what you are referring to is when you've got multiple cases of one variable vs one case of one variable multiplied several times.

If you have loads of observations, the errors tend to sort themselves out. As in, some will deviate to be higher than the mean, but some will deviate to be lower than the mean and the errors cancel. Whereas if you take one observation and multiply it, the error is amplified. So that's where the different formulae come from.

eg 50 observations of bolt sizes will be distributed with a variance of 50σ² whereas 50b, where b is the observation of one bolt size, will be distributed with a variance of 50²σ², IIRC.

Reply 2

Zhy

Original post by Contrad!ction.

eg 50 observations of bolt sizes will be distributed with a variance of 50σ² whereas 50b, where b is the observation of one bolt size, will be distributed with a variance of 50²σ², IIRC.

This is correct. In the exam it's more common to be given the former -- if you are really unsure then just check what your value of the variance is for both. A high variance usually means you've done something wrong.

Reply 3

Contrad!ction.

Original post by Zhy

Ah good. Was trying to come up with a simple example.

Also, OP - normally, you'll use this in conjunction with a hypothesis test, so you could check your end result - does it look right that you're rejecting/accepting the null hypothesis? If it doesn't, then either you've got an odd significance level, or you've done something wrong.