PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Solution. Step 1:
Let V = √ ( k
X/k. Need to find PDF of V . Recall χ 2 k is Gamma 2 , 1 )
so
2
Now,
f X (x) = (1/2)k/2
Γ(k/2) xk/2−1 e −x/2 , for x > 0.
v = h(x) = √ x/k
⇒ h −1 (v) = kv 2
and
h −1 (v) ′ = 2kv
⇒ f V (v) = f X (h −1 (v))|h −1 (v) ′ |
= (1/2)k/2
Γ(k/2) (kv2 ) k/2−1 e −kv2 /2 (2kv)
Step 2:
Apply Theorem 1.8.3 to find PDF of
T = Z ∫ ∞
V = |v|f V (v)f Z (vt) dv
−∞
= 2(k/2)k/2
Γ(k/2) vk−1 e −kv2 /2 , v > 0.
=
∫ ∞
0
v 2(k/2)k/2
Γ(k/2) vk−1 e −kv2 /2 1
√
2π
e −t2 v 2 /2 dv
= (k/2)k/2
Γ(k/2) √ 2π
∫ ∞
0
v k−1 e −1/2(k+t2 )v 2 2v dv;
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