Description

1. If Y = f(Z) be monotonic,
pY(y)dy = pZ(z)dz
2. The pdf of the χ2(k) distribution is given by
x2k−1e−2x (1.1.1)
pX(x) =u(x) (1.2.1)
2 Γ
3. The Beta function is defined as
B(x,y) = (1.3.1)
2 Problems 1. Let X1 ∼ χ2(m) and X2 ∼ χ2(n) be independent. For
show that X1/m
Z = , X2/n (2.1.1)
2. Show that FZ(z) = E FX1 ( )] mzX2
n (2.1.2)
3. Show that pZ(z) = E mX2pX1 ( )] mzX2
n (2.2.1)
pZ(z) = (m(n )xm2−1(1+ mz)−m2+n u(z) (2.3.1) m/2
B, n2 n
Z has an F distribution with (m,n) degrees of freedom.
4. Show that Y = Z1 is monotonic.
5. Show that Y also has an F distribution with (n,m) degrees of freedom.
6. Find the pdf of mZmZ+n.