38.9. INDEPENDENCE AND CONDITIONAL PROBABILITY 791
Proof: From Proposition 38.8.7,(
X−µ
σ
)is normally distributed with mean µ and vari-
ance σ2. If f (t) is the density of(
X−µ
σ
)2, then
F (x)≡∫ x
0f (t)dt ≡ P
((X−µ
σ
)2
< x
)= P
(−√
x <X−µ
σ<√
x)
=1√2π
∫ √x
−√
xe−
12 t2
dt =
√2√π
∫ √x
0e−
12 t2
dt
change variables. Let t2
2 = u so tdt = du,dt = du√2u. Then the above is
√2√
2√
π
∫ x/2
0u−1/2e−udu
Then, taking the derivative will yield the density. This is
1√π
12
( x2
)−1/2e−x/2 =
1√π
1√2√
xe−x/2
=1
Γ(1/2)21/2 x1/2−1e−x/2
because of Corollary 38.1.2, which is the density for X 2 (1) as claimed. ■
38.9 Independence and Conditional ProbabilityIn the above, the concept of Probability that a random variable is in some set has beenconsidered. More generally, you have a set and a collection of subsets of this set and afunction which assigns a number between 0 and 1 to sets in this collection. This is theprobability function. The following has to do with conditional probability.
Definition 38.9.1 Let C be a collection of sets contained in some universal set U. Thesecould be intervals on the real line for example, and U could be R. Let P : C → [0,1]. Thusfor A a set, P(A) ∈ [0,1] . It satisfies the following conditions.
1. If Ai are disjoint sets in C , then P(∪n
i=1Ai)= ∑
ni=1 P(Ai) . More generally, if you
have infinitely many such disjoint sets, P(∪∞i=1Ai) = ∑
∞i=1 P(Ai).
2. If A ∈ C , then P(A)+P(U \A) = 1.
Then one defines the conditional probability as follows. If P(B) ̸= 0,
P(A|B)≡ P(A∩B)P(B)
The sets in C are called events.