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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

\begin{align} F_X(x) &= P_X((-\infty,x])\;\forall x\in \mathbb R \\ &= P(X^{-1}((-\infty, x])) \\ &=P(\{ \omega\in\mathcal S:\;X(\omega)\leq x \}) \end{align}

Notation $\quad$ Normally, we write this as

F_X(x) = P(X\leq x)

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.

\lim_{x\rightarrow\infty}F_X(x) = 1\;\;\mbox{and}\;\;\lim_{x\rightarrow -\infty}F_X(x) = 0

This is because

\lim_{x\rightarrow\infty}F_X(x) = P(\{ \omega\in\mathcal S:\;X(\omega)\leq\infty \})=1

and

\lim_{x\rightarrow -\infty}F_X(x)= P(\varnothing)= 0

2. For any $$ x_1,x_2 $$ ∈ R such that $$ x_1<x_2 $$ ,

F_X(x_1)\leq F_X(x_2)

i.e. $$ F_X(x) $$ is a non decreasing function.

3. $$ F_X $$ is continuous from the right , i.e.

F_X(x^+) \equiv \lim_{\epsilon\rightarrow0,\epsilon>0}F_X(x+\epsilon)=F_X(x)\;\;\forall x\in\mathbb R

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

\lim_{n\rightarrow\infty}A_n = \bigcap_{n=1}^{\infty}A_N

(ii) If $$ A_1 $$ ⊃ $$ A_2 $$ ⊃ ..., then

P(\lim_{n\rightarrow\infty}A_n) = \lim_{n\rightarrow\infty}P(A_n)

(iii) We can write $$ F_X(x^+) $$ as

F_X(x^+) = \lim_{n\rightarrow\infty}F_X(x+\frac{1}{n})

Now let

A = \{X\leq x+\frac{1}{n})\}

Then

\begin{align} F_X(x^+) &= \lim_{n\rightarrow\infty}P(X\leq\frac{1}{n}) \\ &=\lim_{n\rightarrow\infty}P(A_n) \\ &=P(\lim_{n\rightarrow\infty}A_n) \\ &=P(\bigcap_{n=1}^{\infty}A_n) \\ &=P(\bigcap_{n=1}^{\infty}\{X\leq x+\frac{1}{n})\})\\ &=F_X(x) \end{align}

4. $$ P(X>x) = 1-F_X(x) $$ for all x ∈ R

5. If $$ x_1 < x_2 $$ , then

P(x_1<X\leq x_2) = F_X(x_2) - F_X(x_1)\;\forall x_1,x_2\in\mathbb R

6. $P(\{X=x\})= F_X(x) - F_X(x^-)$ , where

F)X(x^-) = \lim_{\epsilon\rightarrow 0,\epsilon>0} F_X(x-\epsilon)

The Probability Density Function

Definition $\quad$ The probability density function (pdf) of a random variable X is the derivative of the cdf of X,

f_X(x) = \frac{dF_X(x)}{dx}

at points where $$ F_x $$ is differentiable.
From the Fundamental Theorem of Calculus, we then have that

F_X(x)=\int_{-\infty}^xf_X(r)dr\;\;\forall x\in\mathbb R

Important note: the cdf $$ F_X $$ might not be differentiable everywhere. At points where $$ F_X $$ is not differentiable, we can use the Dirac delta function to defing $$ f_x $$ .

Definition $\quad$ The Dirac Delta Function $\delta(x)$ is the function satisfying the properties:
1.

\delta(x) = 0 \;\forall x\neq0

2.

\int_{-\infty}^{\infty} \delta(x)dx = \int_{-\epsilon}^{\epsilon}\delta(x)dx = 1\;\forall\epsilon>0

If $$ F_X $$ is not differentiable at a point, use $\delta(x)$ at that point to represent $$ f_X $$ .

Why do we do this? Consider the step function $$ u(x) $$ , which is discontinuous and thus not differentiable at $$ x=0 $$ . This is a common type of discontinuity we see in cdfs. The derivative of $$ u(x) $$ is defined as

\frac{du(x)}{dx}=\lim_{h\rightarrow 0}\frac{u(x+h)-u(x)}{h}

This limit does not exist at $$ x=0 $$

Let's look at the function

g(x) = \frac{u(x+h)-u(h)}{h}

It looks like this:

Fig 1: g(x) for h>0

For any x ≠ 0, we have that

\frac{u(x+h)-u(x)}{h}=0

for small enough h.
Also, ∀ $\epsilon$ <0,

\int_{-\epsilon}^{\epsilon}\delta(x)dx= 1\; \forall h<\epsilon

So, in the limit, the function g(x) has the properties of the $\delta$ -function as h tends to 0. A similar argument can be made for h<0.
So this is why it is sometimes written that

\frac{du(x)}{dx} = \delta(x)

Since we will only work with non-differentiable functions that have step discontinuities as cdfs, we write

f_X(x) = \frac{dF_X(x)}{dx}

with the understanding that $$ d/dx $$ is not necessarily the traditional definition of the derivative.

Properties of the pdf:
1. (proof)

f_X(x)\geq 0\;\forall x\in \mathbb R

2. (proof)

\int_{-\infty}^{\infty}f_X(x)dx = 1

3. (proof) if $$ x_1<x_2 $$ , then

P(x_1<X\leq x_2) = \int_{-x_1}^{x_2}f_X(x)dx

Some notes:

We introduced the concept of a pdf in our discussion of probability spaces. We could have defined the pdf of a random variable X as a function $$ f_X $$ satisfying properties 1 and 2 above, and then define $$ F_X $$ in terms of $$ f_X $$ .
f_X(x) is not a probability for a fixed x, it gives us instead the "probability density", so it must be integrated to give us the probability.
In practice, to compute probabilities of random variable X, we normally use

P(X\in A) = \int_{A}f_X(x)dx

Continuous and Discrete Random Variables

A random variable X having a cdf that is continuous everywhere is called a continuous random variable.

Fig:2 a possible cdf for a continuous random variable

A random variable X having a piece-wise constant cdf is a discrete random variable.

Fig:3 a possible cdf for a discrete random variable

A random variable X whose cdf is neither continuous everywhere nor piece-wise constant is a mixed random variable.

Fig:4 a possible cdf for a mixed random variable

We will consider only discrete and continuous random variables in this course.

Note:

For a continuous random variable X, P(X=x)=0 ∀x ∈ R, because <mathF_X(x)-F_X(x^-) = 0</math> ∀x ∈ R.
We can use the cdf/pdf functions for a discrete random variable X, but generally, we do not. Instead, we use the pmf.

Probability Mass Function

The probability mass function (pmf) of random variable X is the function

P_X(x) \equiv |P_X(\{x\})=P(X=x)

We will use this function when X is discrete.

Although P(X=x) exists for every x ∈ R, we normally define $$ p_X(x) $$ for some subset R⊂R

Definition $\quad$ Given random variable X:S→R, let R = X(S) be the range space of X (note that the range of X is still R).

Example: Let X be the sum of values rolled on two fair die. Then R = {2,3,...,12}.

We define the pmf only on R, so in the example above, we would have $$ P_X(x) = P(X=x) $$ ∀x ∈ R.

We can now consider the probability space for a discrete random variable to be (R,P(R),P $$ _X $$ ).
Note that if X is continuous, we define the cdf/pdf for all x ∈ R. for example, if X = V</math>^2</math>, where V is the voltage (so X is proportional to power), then R=[0,∞), but we still define $$ f_X(x) $$ ∀x ∈ R.

Properties of the pmf:
1. (proof)

P_X(x) \geq 0

2. (proof)

\sum_{x\in\mathcal R}P_X(x) = 1

These properties can be derived from the axioms. Note that we could simple define the pmf to be a function having these two and then create a probability mapping P(.) in such a way that P(.) will satisfy the axioms. We discussed this in our lectures on probability spaces. This is what we do in practice.

References

M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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