Continuous Random Variables

If the random variable $X$ can take values in an interval $[a,b]$ then it is a continuous random variable. To describe the probabilities associated with $X$ we use its probability density function.

$$P(X \in \delta x) = p(x) \delta x$$

The probability $X \in A = \int_{A} p(x) dx$
$$\begin{align} \int_{\mathbb{R}} p(x) dx = 1 \\ \text{i.e.} \int_{-\infty}^{\infty} p(x) dx = 1 \end{align}$$

$$P(X \in [a,b]) = \int_{a}^{b} p(x) dx$$

$$F(a) = P(X < a) = \int_{-\infty}^{a} p(x) dx$$ $F(a)$ is the cumulative density function $\frac{dF}{da} = p(a)$, so differentiating the cumulative distribution function gives the probability density function.

Example 1.
The Uniform Distribution
$$P(X \in [a,b]) = b-a \; 0 \leq a \leq b\leq 1$$
$$\begin{align} p(x) = \begin{cases} 1 & \text{if } x \in [0,1] \\ 0 & \text{otherwise}\\ \end{cases} \end{align}$$ $$\begin{align} F(a) = P(X < a) = \int_0^{a} dx = a \end{align}$$ $$\begin{align} \mathbb{E}[X] = \int_{-\infty}^{\infty} x p(x) dx = [\frac{x^2}{2}]^1_0 = \frac{1}{2} \end{align}$$ $$\begin{align} \mathbb{E}[X^2] = \int_{-\infty}^{\infty} x^2 p(x) dx = [\frac{x^3}{3}]^1_0 = \frac{1}{3} \end{align}$$ $$\begin{align} \text{Var}[X] = \mathbb{E}[X^2] – \mathbb{E}[X]^2 = \frac{1}{3} – (\frac{1}{2})^2 = \frac{1}{12} \end{align}$$