Probability and Probability Distributions

1. Introduction to Probability

Probability is a measure of the likelihood or chance of an event occurring, expressed as a value between 0 (impossible) and 1 (certain).

2. Basic Concepts of Probability

• Classical Definition: If there are $nn$ equally likely outcomes of an event and $mm$ of those outcomes are favorable, the probability of the event is:

$$P(E)=\frac{m}{n}P(E)\; =\; \backslash frac\{m\}\{n\}$$

• Addition Rule: For two events $AA$ and $BB$:

  • If $AA$ and $BB$ are mutually exclusive, $P(A\cup B)=P(A)+P(B)P(A\; \backslash cup\; B)\; =\; P(A)\; +\; P(B)$.

  • If $AA$ and $BB$ are not mutually exclusive, $P(A\cup B)=P(A)+P(B)-P(A\cap B)P(A\; \backslash cup\; B)\; =\; P(A)\; +\; P(B)\; -\; P(A\; \backslash cap\; B)$.

• Multiplication Rule:

  • For independent events $AA$ and $BB$: $P(A\cap B)=P(A)\cdot P(B)P(A\; \backslash cap\; B)\; =\; P(A)\; \backslash cdot\; P(B)$.

  • For dependent events, $P(A\cap B)=P(A)\cdot P(B\mathrm{\mid }A)P(A\; \backslash cap\; B)\; =\; P(A)\; \backslash cdot\; P(B|A)$, where $P(B\mathrm{\mid }A)P(B|A)$ is the conditional probability of $BB$ given $AA$.

3. Probability Distributions

Probability distributions describe how probabilities are distributed over different outcomes.

a. Binomial Distribution

• Definition: Describes the probability of obtaining a fixed number of successes in $nn$ independent Bernoulli trials, each with a success probability $pp$.

• Formula:

$$P(X=k)=\left(\genfrac{}{}{0px}{}{n}{k}\right)p^k(1-p{)}^{n-k}P(X\; =\; k)\; =\; \backslash binom\{n\}\{k\}\; p^k\; (1-p)^\{n-k\}$$

where $\left(\genfrac{}{}{0px}{}{n}{k}\right)\backslash binom\{n\}\{k\}$ is the binomial coefficient.

b. Poisson Distribution

• Definition: Used for modeling the number of events occurring in a fixed interval of time or space when events occur independently.

• Formula:

$$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}P(X\; =\; k)\; =\; \backslash frac\{\backslash lambda^k\; e^\{-\backslash lambda\}\}\{k!\}$$

where $\lambda \backslash lambda$ is the mean number of occurrences.

c. Normal Distribution

• Definition: A continuous probability distribution shaped like a bell curve. It is symmetrical around the mean $\mu \backslash mu$, with standard deviation $\sigma \backslash sigma$.

• Probability Density Function (PDF):

\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{{-\frac{(x-\mu)}2}{2\sigma^2}} \]

4. Expected Value

The expected value (mean) of a random variable is the weighted average of all possible values, where weights are the probabilities of the values. For a discrete random variable $XX$:

$$E(X)=\sum _{i}P(x_i)\cdot x_i$$