I guess is still normally distributed, but what are and ?
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What is 's distribution?
E.g. for : it's "intuitive" to think , but how do you get the answer formally, with procedure? Is it via probability density function ?
Well, CDF is the sum of all the PDFs: .
I'm also looking at Wikipedia on change of variables, but I don't know how to make use of it.
I'm also guessing I shouldn't be doing ?
Actually now this question belongs to the Calculus forum :P
Repeating mr fantastic's words: You should know that where is the c.d.f and is the p.d.f of a random variable X.
So to solve the problem, I am expecting you to know the p.d.f of a normal distribution and chain rule of differentiation. Do you know these?
Isn't what's inside the integral (that I wrote above) "the p.d.f of a normal distribution"?
Notation: I've mistakenly used for .
Chain rule is .
I haven't come upon such a problem before, so I'm not "used to it", as von Neumann would say.
Was I even on the right track in my previous post?
You are on the right track. However you are not using the fact that derivative of a c.d.f is the p.d.f. Your integral along with the term is the c.d.f of a normal distribution, i.e.
Alternatively,
I was telling you how to proceed. You can use the fundamental theorem of calculus to differentiate integrals with respect to a variable that appears in the limit of the integral. In the above expression 'y' appears as a limit of integration. So you have to use the fundamental theorem of calculus.