ECE600 F13 rv Functions of random variable mhossain - Rhea

Random Variables and Signals

Topic 8: Functions of Random Variables

We often do not work with the random variable we observe directly, but with some function of that random variable. So, instead of working with a random variable X, we might instead have some random variable Y=g(X) for some function g:R → R.
In this case, we might model Y directly to bet f $$ _Y $$ (y), especially if we do not know g. Or we might have a model for X and find f $$ _Y $$ (y) (or p $$ _Y $$ (y)) as a function of f $$ _X $$ (or p $$ _X $$ and g.
We will discuss the latter approach here.

More formally, let X be a random variable on (S,F,P) and consider a mapping g:R → R. Then let Y $(\omega)=$ g(X( $\omega))$ ∀ $\omega$ ∈ S.
We normally write this as Y=g(X).

Graphically,

Fig 1: Mapping from S to X to Y under g

Is Y a random variable? We must have Y $^{-1}$ (A) ≡ { $\omega$ ∈ S: Y $(\omega)$ ∈ A} = { $\omega$ ∈ S: g(X $(\omega)$ ) ∈ A} be an element of F ∀A ∈ B(R) (Y must be Borel measurable).
We will only consider functions g in this class for which Y $^{-1}$ (A) ∈ F ∀A ∈ B(R), so that if Y=g(X) for some random variable X, Y will be a random variable.

What is the distribution of Y? Consider 3 cases:

X discrete, Y discrete
X continuous, Y discrete
X continuous, Y continuous

Note: you cannot have a continuous Y from a discrete X.

Case 1: X & Y discrete
Let $$ R_X $$ ≡ X(S) be the range space of X and math>R_Y</math>≡ g(X(S)) be the range space of Y. Then the pmf of Y is

p

$ _Y $

(y) = P(Y=y) = P(g(X)=y)

But this means that

p_Y(y) = \sum_{x\in\mathcal{R}_X:g(x)=y}p_X(x)\;\;\forall y\in\mathcal{R}_Y

Example $\quad$ Let X be the value rolled on a die and

Y = \begin{cases} 1 & \mbox{if}\;X\;\mbox{is odd} \\ 0 & \mbox{if}\;X\;\mbox{is even} \end{cases}

Then R $$ _X $$ = {0,1,2,3,4,5,6} and R $$ _Y $$ = {0,1} and g(x) = x % 2.

Now

p_Y(y) = \sum_{x\in\mathcal{R}_X:g(x)=y}p_X(x)

\begin{align} \Rightarrow p_Y(0) &= p_X(2)+p_X(4)+p_X(6) \\ p_Y(1) &= p_X(1)+p_X(3)+p_X(5) \end{align}

Case 2: X continuous, Y discrete
The pmf of Y in this case is

p

$ _Y $

(y) = P(g(X)=y) = P(X ∈ D</math>_Y</math>) =
where D

$ _Y $

≡ {x ∈ R: g(x)=y} ∀y ∈ R

$ _Y $

ECE600 F13 rv Functions of random variable mhossain - Rhea

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