Hi all, Say I have two random variables X and Y, and also Z=X+Y so if the distribution of these was normal, i would use X+Y-N(mu_x+mu_y, sd_x^2+sd_y^2) so var(Z)=sd_x^2+ sd_y^2 however, say the distribution was poisson X-Poi(lambda_x) Y-Poi(lamda_y) X+Y-Poi(lamda_x+lamda_y) (as is covered in course notes) so then the variance of a poisson distribution is the same as its mean, so i would say that var(Z)=lamda_x+lamda_y, however this isnt the case, by question 12 for the poisson distribtuion we need to use the Var(X+Y)=var(x)+var(y)+2cov(xy) rules for this. My question is, why are we able to just take the variance as given for the normal distribution, but not for the Poisson distribution? would we need to use this variance formula for all distributions except the normal? I hope that makes sense! Just want to clarify so i know when i should be using these variance formulas Thanks Molly
Your formulas for normal and Poisson X+Y hold if X and Y are independent, but not in general. That’s the source of your confusion. Read this page, including the section on correlated variables: https://en.m.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
Yes in Practice Question 4.12 the distributions are not independent, hence the extra covariance term.
