Random Variables and Signals
Topic 6: Random Variables: Distributions
How do we find , compute and model P(x ∈ A) for a random variable X for all A ∈ B(R)? We use three different functions:
- the cumulative distribution function (cdf)
- the probability density function (pdf)
- the probability mass function (pmf)
We will discuss these in this order, although we could come at this discussion in a different way and a different order and arrive at the same place.
Definition $ \quad $ The cumulative distribution function (cdf) of X is defined as
Notation $ \quad $ Normally, we write this as
So $ F_X(x) $ tells us P$ P_X(A) $ if A = (-∞,x] for some real x.
What about other A ∈ B(R)? It can be shown that any A ∈ B(R) can be written as a countable sequence of set operations (unions, intersections, complements) on intervals of the form (-∞,x$ _n $], so can use the probability axioms to find $ P_X(A) $ from $ F_X $ for any A ∈ B(R). This is not how we do things in practice normally. This will be discussed more later.
Can an arbitrary function $ F_X, $ be a valid cdf? No, it cannot.
Properties of a valid cdf:
1.
This is because
and
2. For any $ x_1,x_2 $ ∈ R such that $ x_1<x_2 $,
i.e. $ F_X(x) $ is a non decreasing function.
3. $ F_X $ is continuous from the right , i.e.
Proof:
First, we need some results from analysis and measure theory:
(i) For a sequence of sets, $ A_1, A_2,... $, if $ A_1 $ ⊃ $ A_2 $ ⊃ ..., then
(ii) If $ A_1 $ ⊃ $ A_2 $ ⊃ ..., then
(iii) We can write $ F_X(x^+) $ as
Now let
Then
References
- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.
