Hello,

for the format, look here : http://www.mathhelpforum.com/math-help/latex-help/, the threads in the upper part...

as for your problem :

find the distribution ofIf X1,...,Xk are independent N(0,1) random variables, then

W = (X1)^2 + (X2)^2 + ... + (Xk)^2

has a Chi-squaredk distribution, where k is the degrees of freedom (number of independent squares in the sum of W)

the pdf of N(0,1) is , for t in

now this is a method (law of the unconscious statistician) i can't stop to do. otherwise (it's very similar) you just have to use the jacobian transformation of the pdf. (a search in this subforum may give you some results)

remember that the jacobian transformation has to be a diffeomorphism (in other words a bijection)

for any measurable function h,

if you make right now the jacobian transformation y=x², you'll have a problem since it's not a bijection.

so just note that the integrand is an even function.

hence

now make the transformation y=x². this gives

--> (that's simple calculus & manipulations)

so the pdf of is , which is the pdf of a Gamma distribution

(there are several conventions for the parameters of a gamma distribution, so it may be (2,1/2))

for here, the most straigthforward method is to use the mgf of a gamma distribution.If Z1 ~ Gamma (alpha1, beta) and Z2 ~ Gamma (alpha2, beta); Z1 and Z2 are independent, then Z = Z1 + Z2 ~ Gamma(alpha1+alpha2, beta).

and to remember that the mgf of the sum is the product of the mgf.

then use this last part to conclude that W follows a chi square distribution, by noting that a chi square distribution is a gamma distribution