Summary of Statistics Exam: Time Series and Probability
Statistics Exam: Time Series and Probability Study Guide
Introduction
Probability studies how likely events are to occur. It provides tools to quantify uncertainty and to model random processes such as tossing coins, rolling dice, or drawing cards. This material focuses on core probability concepts, rules for combining probabilities, and practical examples you can use without requiring prior knowledge of statistics or the standard normal distribution.
Basic Concepts
Sample space and outcomes
Definition: The sample space $S$ is the set of all possible outcomes of a random experiment.
- Example: For a fair coin, $S = {\text{heads},\text{tails}}$.
- Example: For a fair six-sided die, $S = {1,2,3,4,5,6}$.
- When experiments are combined, outcomes are paired: for a coin and a die tossed together the sample space is $S = {\text{heads},\text{tails}}\times{1,2,3,4,5,6}$ and has $2\times 6 = 12$ outcomes.
Events
Definition: An event is any subset of the sample space. If $A$ is an event, $A\subseteq S$.
- Single outcome event: ${\text{heads}}$.
- Compound event: "roll an even number" $={2,4,6}$.
Probability of an event
Definition: For a finite sample space with equally likely outcomes, the probability of an event $A$ is $P(A) = \dfrac{|A|}{|S|}$ where $|\cdot|$ denotes the number of outcomes.
- Example: Probability of rolling a 4 on a fair die: $P({4}) = \dfrac{1}{6}$.
- Example: Probability of getting heads and a 1 when tossing a coin and die: $P({(\text{heads},1)})=\dfrac{1}{12}$.
Rules of Probability
Basic rules
- $0 \le P(A) \le 1$ for any event $A$.
- $P(S) = 1$ and $P(\varnothing) = 0$.
- Complement rule: $P(A^c) = 1 - P(A)$.
Definition: Events $A$ and $B$ are mutually exclusive if $A\cap B = \varnothing$.
- If $A$ and $B$ are mutually exclusive then $P(A\cup B) = P(A) + P(B)$.
General addition rule
For any two events $A$ and $B$: $$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$
- Use this when events are not mutually exclusive.
Multiplication rule and independence
Definition: Events $A$ and $B$ are independent if $P(A\cap B) = P(A)P(B)$.
- If events are independent, then $$P(A\cap B) = P(A)P(B).$$
- For dependent events, use conditional probability.
Conditional probability
Definition: The conditional probability of $A$ given $B$ is $$P(A\mid B) = \dfrac{P(A\cap B)}{P(B)},\quad\text{provided }P(B)>0.$$
- Rearranged: $P(A\cap B) = P(A\mid B)P(B)$.
- Example: If a deck had special known composition, and you know the first draw was a red card, probabilities for the second draw may change.
Counting Principles (useful before computing probabilities)
Fundamental counting rule
- If one experiment has $m$ outcomes and another independent experiment has $n$ outcomes then combined there are $m\times n$ outcomes.
Permutations and combinations (brief)
- Permutations (ordered selections): number of ways to arrange $r$ items from $n$ is $$P(n,r) = \dfrac{n!}{(n-r)!}.$$
- Combinations (unordered selections): number of ways to choose $r$ items from $n$ is $$C(n,r) = \dfrac{n!}{r!(n-r)!}.$$
Table: Permutations vs Combinations
| Situation | Order matters? | Formula |
|---|---|---|
| Permutations | Yes | $P(n,r)=\dfrac{n!}{(n-r)!}$ |
| Combinations | No | $C(n,r)=\dfrac{n!}{r!(n-r)!}$ |
Common Probability Distributions (overview)
- Discrete uniform: finite outcomes equally likely, e.g. fair die.
- Binomial: counts successes in $n$ independent trials with success probability $p$; probability of $k$ successes is $$P(X=k)=C(n,k)p^k(1-p)^{n-k}.$$
- Geometric: number of trials until first success.
- Poisson: models counts in fixed intervals with rate $\lambda$.
Table: Discrete distribution quick facts
| Distribution | Support | Key parameter(s) |
|---|---|---|
| Uniform (discrete) | finite set | equally likely outcomes |
| Binomial | $k=0,1,\dots,n$ | $n$, $p$ |
| Poisson | $k=0,1,2,\dots$ | $\lambda$ |
Practical Examples and Applications
Example 1: Coin and die
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Basic Probability Concepts
Klíčová slova: Statistics, Probability, Time Series, Standard Normal Distribution
Klíčové pojmy: Sample space S is set of all outcomes, For equally likely outcomes P(A)=|A|/|S|, Complement rule: P(A^c)=1-P(A), Addition rule: P(A∪B)=P(A)+P(B)-P(A∩B), Independence: P(A∩B)=P(A)P(B), Conditional: P(A|B)=P(A∩B)/P(B), Fundamental counting: m and n outcomes ⇒ m×n outcomes, Binomial: P(X=k)=C(n,k)p^k(1-p)^{n-k}