I am a bit confused with the notation, Whenever I see N(mu,alpha) does that mean the data set is normal and does it also mean that it's standard normal? Is there a difference between using Z~N(mu,alpha) vs. X~N(mu,alpha)? does the Z indicate standard normal?, if so why don't we just use Z(mu,alpha)? Can I still use this notation N(_,_) when the data set is NOT normal? THANKS IN ADVANCE :D
Standard notation: X~N(m,d) means X is a random variable, with a normal distribution having a mean m and standard deviation d. If you wrote Z(m,d) then you need to define Z. N(m,d) means the distribution is normal - you can't use it for something else - it is a matter of notation. If d is unknown for a normal distribution with a known m, then, as you observed, you can specify the midpoint of the distribution, but nothing else.