MATH 350 Notes for Exams

• Independence of $A,B \subset S$: $\mathbb{P}[A \cap B] = \mathbb{P}[A] \mathbb{P}[B]$
• Binomial density: $f(x | K, p) = { K \choose x} p^x (1-p)^{K - x}$
• Bernoulli density: $f(x | p) = p^x (1-p)^{1 - x}$
• Normal density: $f(x | \mu, \sigma) = (2\pi\sigma^2)^{-1/2}e^{-(x - \mu)^2 / (2\sigma^2)}$
• Exponential density: $f(x | \lambda) = \lambda e^{-\lambda x}$
• Continuous Uniform density: $f(x | a, b) = 1 / (b - a)$
• Poisson density: $f(x | \lambda) = e^{-\lambda } \lambda^x / x!$
• Cumulative Distribution Function: $F(x) = \mathbb{P}[X \leq x]$
• Probability: $\mathbb{P}[A] = \sum_{x \in A} f(x | \theta)$
• Expectation of $X$: $\mathbb{E}[X] = \sum_{x \in S} x * f(x | \theta)$
• Discrete Uniform density: $f(x | a, b) = 1 / (b - a + 1)$
• Expectation Binomial: $\mathbb{E}[X] = K * p$
• Variance Binomial: $\mathbb{V}[X] = K * p * (1 - p)$
• General Expectation: $\mathbb{E}[g(X)] = \int f(x) g(x) dx$ for density function $f(x)$ and arbitrary function $g(x)$
• Variance: $\mathbb{V}[X]$ is defined by a general expectation with $g(x) = (x - \mathbb{E}[X])^2$
• Probability: $\mathbb{P}[X \in A]$ is defined by a general expectation with $g(x) = 1_A(x)$