PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
⎧∑
h(x)p(x)
⎪⎨ x
E{h(X)} =
⎪⎩
∫ ∞
−∞
h(x)f(x) dx
if ∑ |h(x)|p(x) < ∞ (X discrete)
x
if
∫ ∞
−∞
|h(x)|f(x) dx < ∞ (X continuous)
The moment generating function of a RV X is defined to be:
M X (t) = E[e tX ] =
↓
moment
generating fn
of RV X.
⎧∑
e tx p(x)
x ⎪⎨
⎪⎩
∫ ∞
−∞
e tx f(x) dx
X discrete
X continuous
M X (0) = 1 always; the mgf may or may not be defined for other values of t.
If M X (t) defined for all t in some open interval containing 0, then:
1. Moments of all orders exist;
2. E[X r ] = M (r)
X
(0) (rth order derivative) ;
3. M X (t) uniquely determines the distribution of X:
M ′ (0) = E(X)
M ′′ (0) = E(X 2 ),
and so on.
1.1 Discrete distributions
1.1.1 Bernoulli distribution
Parameter: 0 ≤ p ≤ 1
Possible values: {0, 1}
Prob. function:
⎧
⎪⎨ p, x = 1
p(x) =
⎪⎩
1 − p, x = 0
2
= p x (1 − p) 1−x