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Revision as of 19:49, 23 November 2013
Random Variables and Signals
Topic 10: Characteristic Functions
Characteristic Functions
The pdf f$ _X $ of a random variable X is a function of a real valued variable x. It is sometimes useful to work with a "frequency domain" representation of f$ _X $. The characteristic function gives us this representation.
Definition $ \qquad $ Z:S → C defined on (S,F,P) is a complex random variable if
where X and Y are real valued random variables on (S,F,P).
Using the linearity of E[],
Now consider the complex random variable Z = $ e^{i\omega X} $, where $ \omega $ ∈ R is a "frequency" variable. We can write Z as
and
This expectation depends on $ \omega $ ∈ R and will be the characteristic function of X.
Definition $ \qquad $ Let X be a random variable on (S,F,P). The characteristic function X is given by
If X is continuous, we have
And if X is discrete, then we use
Note: The characteristic function looks like the Fourier Transform of f$ _X $, with opposite sign in the exponent. We can show that
Moments
Definition $ \qquad $ The Moment Generating Function (mgf) of random variable X is given by
Moment Theorem $ \qquad $ The Moment Theorem (or Moment Generating Theorem) shows us how to use the mgf to find moments of X:
given a random variable X with mgf $ \phi_X $, the nth moment pf X is given by
Proof:
Differentiating $ \phi_X $ with respect to s n times gives
So,
This result can be written in terms of the characteristic function:
Example $ \qquad $ X is an exponential random variable. We can show that
since
Now,
and
So,
and
Then
References
- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.
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