Summary of Statistics: Time Series and Probability
Statistics: Time Series & Probability Explained for Students
Introduction
Statistics is the study of collecting, summarising, analysing and interpreting data to make informed decisions. This guide covers core probability and distribution ideas, sampling and the normal distribution, hypothesis-related formulas, and practical examples to build intuition.
Definition: Statistics is the discipline that deals with methods for collecting, organising, presenting, analysing and interpreting data.
1. Probability basics
Sample space and outcomes
- The sample space $S$ is the set of all possible outcomes of an experiment.
- If you toss a coin and roll a six-sided die together, each paired outcome is of the form $( ext{coin}, ext{die})$ where coin $\in{H, T}$ and die $\in{1,2,3,4,5,6}$.
Example: For the coin-and-die experiment the sample space size is $|S| = 2 \times 6 = 12$.
Basic probability rules
- For equally likely outcomes: $P(A) = \dfrac{|A|}{|S|}$.
- Probabilities range from $0$ to $1$ and $P(S)=1$.
- For complementary events: $P(A^c)=1-P(A)$.
Definition: An event is a subset of the sample space; an outcome is an element of the sample space.
2. Discrete random variables and distributions
- A discrete random variable takes countable values (e.g., number of tails in three coin tosses).
- The probability mass function (pmf) gives $P(X=x)$ for each possible $x$.
Example: Let $X$ be the number of tails in three unbiased coin tosses. Possible values: $0,1,2,3$. The pmf: $P(X=0)=\dfrac{1}{8}$, $P(X=1)=\dfrac{3}{8}$, $P(X=2)=\dfrac{3}{8}$, $P(X=3)=\dfrac{1}{8}$.
3. The Normal distribution and standard normal
- A continuous random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$; we write $X\sim N(\mu,\sigma^2)$.
- The standard normal distribution is $Z\sim N(0,1)$.
Definition: The standardisation formula is $Z = \dfrac{X-\mu}{\sigma}$, which transforms $X\sim N(\mu,\sigma^2)$ to $Z\sim N(0,1)$.
Using standard normal tables or calculator
- To compute $P(a\le X\le b)$ when $X\sim N(\mu,\sigma^2)$ use $$P(a\le X\le b) = P\left(Z \le \dfrac{b-\mu}{\sigma}\right) - P\left(Z \le \dfrac{a-\mu}{\sigma}\right).$$
- For a single tail: $P(X<k)=P\left(Z<\dfrac{k-\mu}{\sigma}\right)$.
Example 1: If lightbulb life $X\sim N(900,35^2)$, probability that a bulb lasts less than 950 hours: Compute $z=\dfrac{950-900}{35}=\dfrac{50}{35}\approx 1.4286$. Then $$P(X<950)=P(Z<1.4286)\approx 0.9236.$$ So about $92.36%$ of bulbs last less than 950 hours.
Example 2: Probability a bulb lasts between 930 and 940 hours when $X\sim N(900,35^2)$: Compute $z_1=\dfrac{930-900}{35}=\dfrac{30}{35}\approx 0.8571$, $z_2=\dfrac{940-900}{35}=\dfrac{40}{35}\approx 1.1429$. Then $$P(930<X<940)=P(Z<1.1429)-P(Z<0.8571)\approx 0.873 - 0.804\approx 0.069.$$ (Use table or calculator for precise value.)
Example 3: For a sample mean. If $X_i$ are iid with mean $\mu$ and sd $\sigma$, the sample mean $\overline{X}$ of size $n$ has mean $\mu$ and sd $\sigma/\sqrt{n}$. So $$Z=\dfrac{\overline{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)\text{ approximately (for large }n\text{).}$$
4. Central Limit Theorem (CLT)
- The CLT states: Regardless of the population distribution, the distribution of the sample mean $\overline{X}$ is approximately normal when the sample size $n$ is large.
💡 Did you know?Fun fact: The CLT explains why normal approximations work so well in many practical situations even when the underlying data aren't normally distributed.
5. Standard procedures and formulas (useful at glance)
- Sample mean: $$\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n} x_i.$$
- Sample variance: $$s^2=\dfrac{1}{n-1}\sum_{i=1}^{n} (x_i-\overline{x})^2.$$
- Standardisation for single value: $$Z=\dfrac{X-\mu}{\sigma}.$$
- Standardisation for sample mean: $$Z=\dfrac{\overline{X}-\mu}{\sigma/\sqrt{n}}.$$
Table: Comparison of single observation vs sample mean
| Quantity | Mean | Standard deviation | Use |
|---|---|---|---|
| Single observation $X$ | $\mu$ | $\sigma$ | Probabili |
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Basic Statistics Essentials
Klíčová slova: Time Series Analysis, Statistics
Klíčové pojmy: Sample space size is product of independent component counts, For equally likely outcomes: P(A)=|A|/|S|, Standardise: Z=(X-μ)/σ to use standard normal, Sample mean sd is σ/√n, CLT: sample mean approx normal for large n, Use table lookup to convert P(Z<k) to k, Number of tails in 3 tosses pmf: 1/8,3/8,3/8,1/8, Probability for range: P(a≤X≤b)=Φ((b-μ)/σ)-Φ((a-μ)/σ)