Do you know the mgf of a chi square random variable?
Do you know the mgf of the sum of two random variables?
Let X1 and X2 be two independent random variables.
Let X1 and Y = X1 + X2 be χ2(r1) (Chi Square) and χ2(r),
respectively, where r1 < r.
(a) Find the mgf of X2.
(b) What is its distribution?
Do you know the mgf of a chi square random variable?
Do you know the mgf of the sum of two random variables?
Chi-squared or χ2
: Let Z1, Z2, · · · , Zn be independent standard normal
RV’s. Let X = Sum Zi^2 i=1...n
Then X has the chi-squared distribution with n degrees of freedom. It can
be shown that this is the gamma distribution with α = n/2 and β = 1/2. So
the pdf is
f(x|n) = 1/(2^(n/2)(Γ(n/2)) x^(n/2−1)e^(−x/2), x ≥ 0
E[X] = n, V ar(X) = 2n, M(t) = (1/(1-2t))^(n/2)
Note that the sum of independent chi-squared is again with chi-squared with
the number of degree of freedom adding.
aytajalli (02-01-2015)
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