2.6 Continuous Random Variables

A variable that can take any value within a range, requiring calculus to describe its probabilities. 📈

Continuous Random Variable

A continuous random variable is a mapping \(X\) of all the events of a sample space to numerical values \(x \in \mathbb{R}\). \[ X: Event \in Sample \ Space \to x \in \mathbb{R} \]

Probability Density Function (PDF)

A probability density function (PDF) is a mapping of values \(x\) of intervals of continuous random variables \(X\) to probabilities \([0,1]\).

\[ P(a \leq X \leq b) = \int_{a}^{b} f_{X}(x) \ dx \]

PDF has properties: \[ f_{X}(x) \geq 0, \ \int_{-\infty}^{\infty} f_{X}(x) \ dx = 1 \]

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for continuous random variables is defined as:

\[ F_{X}(x) = P(X \leq x) = \int_{-\infty}^{x} f_{X}(u) \ du \]

Relation to PDF:

\[ f_{X}(x) = \frac{dF_{X}(x)}{dx} \] \[ F_{X}(x) = \int_{-\infty}^{x} f_{X}(u) \ du \]

Illustration: PDF and CDF

Expectation and Variance of PDFs

Continuous expectation, or mean, is the average numerical value that the continuous random variable takes over the PDF. \[ E[X] = \int_{-\infty}^{\infty} x \ f_{X}(x) dx \]

Continuous variance is the expected squared difference from the mean of a PDF. \[ \text{Var}[X] = E[(X - E[X])^{2}] = \int_{-\infty}^{\infty} (x - E[X])^2 \ f_{X}(x) dx \]

Joint and Marginal PDFs

The joint PDF calculates the intersection of two continuous random variables: \[ P((X,Y) \in A) = \int \int_{A} f_{X,Y}(x_{1}, x_{2}) dx_{1} \ dx_{2} \]

From the previous definition we can also compute the marginal PDF for a particular random variable:

\[ f_{X}(x_{1}) = \int_{-\infty}^{\infty} f_{X,Y}(x_{1}, x_{2}) dx_{2} \]

\[ f_{Y}(x_{2}) = \int_{-\infty}^{\infty} f_{X,Y}(x_{1},x_{2}) dx_{1} \]

Conditional Expectation of PDFs

The conditional PDF gives the probability distribution of a continuous random variable given that another variable has a specific value. \[ f_{X|Y}(x_{1}| x_{2}) = \frac{f_{X,Y}(x_{1}, x_{2})}{f_{Y}(x_{2})} \]

The conditional expectation is the expected value of a continuous random variable given that another variable is fixed at a specific value. \[ \mathbb{E}[X| Y = x_{2}] = \int_{x_{1} \in X} x_{1} \frac{f_{X,Y}(x_{1}, x_{2})}{f_{Y}(x_{2})} dx_{1} \]

Independence

Two continuous random variables are independent if: \[ f_{X,Y}(x_{1}, x_{2}) = f_{X}(x_{1}) f_{Y}(x_{2}) \]

This can be extended to multiple random variables: \[ f_{X_{1},...,X{n}}(x_{1}, ..., x_{n}) = f_{X_{1}}(x_{1}) ... \ f_{X_{n}}(x_{n}) \]