Continuous Random Variables

Learning Outcomes

• Continuous Random Variables

  • Probability Density Functions

  • Cumulative Density/Distribution Function

• Common Distributions

Continuous Random Variables

Continuous Random Variable

A random variable \(X\) is considered continuous if the \(P(X=x)\) does not exist.

Distribution	Parameters	PDF
Uniform	\(a\) and \(b\)	\(\frac{1}{b-a}\)
Normal	\(\mu\) and \(\sigma^2\)	\(\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}\)
Exponential	\(\lambda\)	\(\lambda e^{-\lambda x}\)
Gamma	\(\alpha\) and \(\beta\)	\(\frac{x^{\alpha-1}e^{-x/\beta}}{\beta^\alpha\Gamma(\alpha)}\)
Beta	\(\alpha\) and \(\beta\)	\(\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int^1_0x^{\alpha-1}(1-x)^{\beta-1}dx}\)

Cumulative Density Function

The cumulative distribution function of \(X\) provides the \(P(X\leq x)\), denoted by \(F(x)\), for the domain of \(X\).

Properties of the CDF of \(X\):

1. \(F(-\infty)\equiv \lim_{y\rightarrow -\infty}F(y)=0\)
2. \(F(\infty)\equiv \lim_{y\rightarrow \infty}F(y)=1\)
3. \(F(x)\) is a nondecreaseing function

Probability Density Function

The probability density function of the random variable \(X\) is given by

\[ f(x)=\frac{dF(x)}{d(x)}=F^\prime(x) \]

wherever the derivative exists.

Properties of pdfs:

1. \(f(x)\geq 0\)
2. \(\int^\infty_{-\infty}f(x)dx=1\)
3. \(P(a\leq X\leq b) = P(a<X<b)=\int^b_af(x)dx\)

PDF to CDF

Obtaining Probability

Uniform Distribution

Uniform Distribution

A random variable is said to follow uniform distribution if the density function is constant between two parameters.

PDF

CDF

Exponential Distribution

Exponential Distribution

An exponential distribution is used to model positive continuous random variables, commonly used for time.

PDF

CDF

Normal Distribution

Normal Distribution

A random variable is said to follow a normal distribution if the the frequency of occurrence follow a Gaussian function.

PDF

Gamma Distribution

Gamma Distribution

A gamma distribution is the general form of distribution for a \(\chi^2\)-distribution and exponential distribution.

PDF

Beta Distribution

Beta Distribution

A beta distribution models a continuous random variable that only has support between the 0 and 1.

PDF