Probability & Distributions | Binomial Poisson Normal

Module IV: Statistical Techniques – II – All Important Formulas

Addition Law

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Multiplication Law

P(A \cap B) = P(A)\,P(B|A) = P(B)\,P(A|B)

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0

Bayes’ Theorem

P(A_i|B) = \frac{P(B|A_i)\,P(A_i)}{\sum_{j} P(B|A_j)\,P(A_j)}

Discrete Random Variable – Probability Mass Function (PMF)

p(x_i) = P(X = x_i), \qquad \sum_i p(x_i) = 1

Continuous Random Variable – Probability Density Function (PDF)

f(x) \ge 0, \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1

P(a \le X \le b) = \int_a^b f(x)\,dx

Cumulative Distribution Function (CDF)

F(x) = P(X \le x)

Expectation

E(X) = \begin{cases} \sum x_i\,p(x_i) & \text{(discrete)} \\[1mm] \int_{-\infty}^{\infty} x\,f(x)\,dx & \text{(continuous)} \end{cases}

Variance

\operatorname{Var}(X) = E(X^2) - [E(X)]^2

E(X^2) = \begin{cases} \sum x_i^2\,p(x_i) & \text{(discrete)} \\[1mm] \int_{-\infty}^{\infty} x^2\,f(x)\,dx & \text{(continuous)} \end{cases}

Standard Deviation

\sigma = \sqrt{\operatorname{Var}(X)}

P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x=0,1,2,\dots,n

E(X) = np, \qquad \operatorname{Var}(X) = np(1-p)

\text{MGF: } M(t) = (q + pe^t)^n, \quad q=1-p

P(X=x) = e^{-\lambda} \frac{\lambda^x}{x!}, \quad x=0,1,2,\dots

E(X) = \lambda, \qquad \operatorname{Var}(X) = \lambda

\text{MGF: } M(t) = e^{\lambda(e^t-1)}

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right), \quad -\infty < x < \infty

E(X) = \mu, \qquad \operatorname{Var}(X) = \sigma^2

Standard Normal Variate

Z = \frac{X - \mu}{\sigma} \sim N(0,1)

\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}

Properties