Cauchy Density
The Cauchy distribution is a continuous probability distribution that is defined by its probability density function (PDF): $$p(x) = \frac{1}{\pi (1 + x^2)}$$
Expectation and Variance of Continuous Random Variable
Example
A random variable X has the pdf $$p(x) = \left\{ \begin{array}{ll} \frac{3}{40} (8 - x^2); & 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{array} \right.$$ Check that this is a valid pdf and compute the expected value and the variance of this random variable.
Uniform Distribution
Interactive Uniform Distribution
Adjust parameters \(a\) and \(b\).
PDF \(p(x)\)
CDF \(F(x)\)
Parameter a: 0.00
Parameter b: 1.00
Example
A random variable X is uniformly distributed on the interval $[a, b]$. Compute the expected value and the variance of this random variable.
Uniform Distribution expectation and variance
Example
A random variable X is uniformly distributed on the interval $[-1, 4]$
Exponential Distribution
Interactive Exponential Distribution
Adjust parameter \(\lambda\).
PDF \(p(x) = \lambda e^{-\lambda x}\)
CDF \(F(x) = 1 - e^{-\lambda x}\)
Parameter \(\lambda\): 1.00
Example
A random variable X is exponentially distributed with parameter $\lambda = 1$. Compute the expected value and the variance of this random variable.
Exponential Distribution expectation and variance
Example
A random variable X is exponentially distributed with parameter $\lambda = 2$.
Exponential Distribution cdf
Normal Distribution
Interactive Gaussian Distribution
Adjust parameters \(\mu\) and \(\sigma\).
PDF \(p(x)\)
CDF \(F(x)\)
Parameter \(\mu\): 0.00
Parameter \(\sigma\): 1.00
Example
A random variable X is standard normal, i.e. Gaussian with parameters $\mu = 0$ and $\sigma = 1$. Compute the expected value and the variance of this random variable.
Gaussian Distribution expectation and variance
Standard Gaussian CDF
Theorem:
Central Limit Theorem
Let $X_1, X_2, \dots, X_n$ be a sequence of independent and identically distributed random variables with finite mean $\mu$ and finite variance $\sigma^2>0$. Then the average $$\bar X = \frac{X_1 + X_2 + \dots + X_n}{n}$$ is approximately normally distributed with mean $\mu$ and variance $\sigma^2/n$ as $n \to \infty$.