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Probability density function
\begin{eqnarray}
P(a \leq X \leq b) = \int_a^b f(x)dx \\
f(x) \geq 0 \ \ (- \infty \leq x \leq \infty) \\
\int_{-\infty}^{\infty} f(x)dx = 1
\end{eqnarray}
Cumulative distribution function
\begin{eqnarray}
F(x) = \int_{-\infty}^x(fu)du \\
\frac{d}{dx}F(x) = f(x)
\end{eqnarray}
Expected value
\begin{eqnarray}
E(x) = \int_{-\infty}^{\infty}xf(x)dx = \mu \\
\end{eqnarray}
Variance
\begin{eqnarray}
V(x) = E \{ (X - \mu ) \}^2 = \int_{-\infty}^{\infty}(x - \mu)^2f(x)dx = \sigma^2
\end{eqnarray}
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