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MAST20006/MAST90057 – Module 2. Discrete Distributions

创建日期：2024-05-23 16:17:19

所属分类：数学mathematics

标签： MAST20006

Discrete Distributions

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MAST20006/MAST90057 – Module 2. Discrete Distributions

Module 2. Discrete Distributions
Chapter 2 in the textbook
1/88
MAST20006/MAST90057 – Module 2. Discrete Distributions
Overview
1 Discrete random variables
2 Mathematical expectation
3 Mean, variance and standard deviation
4 Bernoulli trials and the binomial distribution
5 The moment-generating function
6 The Poisson distribution
2/88
MAST20006/MAST90057 – Module 2. Discrete Distributions
Discrete random variables
1. Discrete random variables
Recall that a fundamental objective of probability theory is to find
the probability of a given event B in the sample (outcome) space S.
It can be difficult to describe and analyse S, and accordingly B, if
the elements of S are not numerical.
However, one often deals with situations where one can associate
with each sample point (outcome) s in S a numerical measurement
x ; that makes life easier.
The numeric measurement x, when regarded as a function of sample
point s, is called a random variable, and is denoted as X or X(s).
3/88
MAST20006/MAST90057 – Module 2. Discrete Distributions
Discrete random variables
Definition 1
Given a random experiment with an outcome space S, a function X that
assigns to each element s in S a real number X(s) = x is called a
random variable (abbr. r.v.).
The range (or space) of X is the set of real numbers
{x : X(s) = x, s ∈ S}, where ‘s ∈ S’ means the element s belongs to the
set S.
Remarks : The range of X is often denoted as X(S) or SX .
Now each event (subset) B in S can be described by the subset
A := X(B) of real numbers assumed by some function (r.v.) X on
B.
Note that A is a subset of SX but not of S, and that X(B) does
not specify B for a general X.
4/88
MAST20006/MAST90057 – Module 2. Discrete Distributions
Discrete random variables
So X : S → SX ⊆ R, such that s 7→ X(s) = x, and for every
A ⊆ SX , there exists B ⊆ S such that
A = X(B) = {x : x = X(s), s ∈ B}
and therefore B = {s ∈ S : X(s) ∈ A}.
Namely, for A ⊆ SX ,
PX(A) = P (X ∈ A) = P ({s ∈ S : X(s) ∈ A}) = P (B)
In particular,
PX(SX) = P (X ∈ SX) = P ({s : s ∈ S : X(s) ∈ SX}) = P (S) = 1.
i.e. the probability of the range of X equals 1.
5/88
MAST20006/MAST90057 – Module 2. Discrete Distributions
Discrete random variables
Assigning probability to A = X(B) ∈ SX can be easier than
assigning probability to B ∈ S, as A is of numerical nature, while B
is not necessarily numerical.
Difficulties still remain :
1 How to assign a probability to a subset A = X(B) ∈ SX ?
2 How to define a r.v. X as a function of s ∈ S ?
The response to 2) is determined by the problem under
consideration, and is not unique.
To answer 1) we will focus on the discrete sample space at this
stage.
If S is discrete, SX is also discrete. So we would be able to calculate
PX(A) for any subset A in SX if we have assigned a probability
for each element in SX .
(Remember there exists a B ⊆ S such that A = X(B).)

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