For an infinitesimally small interval of width \(\delta x\),

\[ P(x \lt X \lt x+\delta x) \;\approx\; f(x) \times \delta x\]

If the whole range of possible x-values is split into such slices, the definition of an expected value for a discrete random variables would give

\[ E[X] \;\approx\; \sum {x \times f(x) \; \delta x}\]

In the limit, this summation becomes an integral, giving us the following definition.

Definition

The expected value of a continuous random variable with probability density function \(f(x)\) is

\[ E[X] \;=\; \int_{-\infty}^{\infty} {x \times f(x) \; d x}\]

This can be generalised:

Definition

If \(X\) is a continuous random variable with probability density function \(f(x)\), the expected value of any function \(g(X)\) is

\[ E\big[g(X)\big] \;=\; \int_{-\infty}^{\infty} {g(x) \times f(x) \; d x}\]