PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Figure 16: Discrete case
Figure 17: Normal approximation to binomial (an application of the Central Limit
Theorem) continuous case
Remarks
⎧
⎪⎨ 0 x < α
1. If we take F (x) =
⎪⎩
1 x ≥ α
then we have X = α with probability 1, and convergence in distribution to this
F is the same thing as convergence in probability to α.
2. Commonly used notation for convergence in distribution is either:
(i) L[X n ] → L[X]
(ii) X n −→
D
L[X] or e.g., X n −→
D
N(0, 1).
3. An important result that we will use without proof is as follows:
Let M n (t) be MGF of X n and M(t) be MGF of X. Then if M n (t) → M(t) for
each t in some open interval containing 0, as n → ∞, then L[X n ] −→
D
L[X].
(Sometimes called the Continuity Theorem.)
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