Expectation of a continuous random variable
Expectation of a continuous random variable¶
Let \(X\) be a discrete random variable taking real values. The expectation of \(X\) is defined by:
where \(p(x)\) is the PDF of \(X\).
Example: The expectation of a uniform random variable¶
Take a uniform random variable:
Its expectation is:
which makes a lot of sense because it is the point between \(a\) and \(b\).
Interpretation of the expectation of continuous random variables¶
Mathematically, the expectation is the x-coordinate of the centroid of the area between the x-axis and the PDF, see Fig. 9.
As in the discrete case, you can think of the expectation as the value one should “expect” to get. But, again, take this interpretation with a grain of salt… See Fig. 10 for an example where the expectation is a very improbable value.
In Fig. 10, the PDF of the random variable has two distinct peaks. Each peak of the PDF is called a mode. “Nice” PDF’s have a single mode. They are called unimodal. For unimodal PDF’s the interpretation that the expectation is the value you should expect to get makes a lot of sense. However, a PDF may be multi-modal. Then the expectation is not very useful…