Ch 3 Probability

Random Variables & Probability Distributions

Why Variation Appears

• Almost every physical or engineering experiment shows random variation in its outcomes. :contentReference[oaicite:0]{index=0}
  • Measurements rarely repeat exactly because of hidden environmental factors, instrument noise, or truly random processes. :contentReference[oaicite:1]{index=1}
• Such experiments are called random experiments; classic examples include rolling dice or flipping coins. :contentReference[oaicite:2]{index=2}

Probability vs Statistics

• Probability starts with a known model of the process and deduces the likelihood of future events. :contentReference[oaicite:3]{index=3}
  • Developed historically to analyse games of chance and to manage risk in engineering design. :contentReference[oaicite:4]{index=4}
• Statistics begins with observed data and infers an underlying probability model, then uses that model to make decisions or predictions. :contentReference[oaicite:5]{index=5}
  • Bayesian statistics refines an initial (prior) probability $P(\theta )$ with new evidence to obtain an updated (posterior) distribution. :contentReference[oaicite:6]{index=6}

Random Variables

• A random variable $X$ assigns a real-number value to each outcome of a random experiment. :contentReference[oaicite:7]{index=7}
  • Discrete: takes countable values (e.g., $\{0,1,2\}$ heads).
  • Continuous: takes any value in an interval (e.g., temperature).
  • Each possible value comes with a probability; together they form a probability distribution. :contentReference[oaicite:8]{index=8}

Uncertainty in Experiments

• Random errors arise from unpredictable fluctuations (noise, drift); systematic errors shift all readings the same way. :contentReference[oaicite:9]{index=9}
• Always report measurements as value $\pm$ uncertainty and propagate uncertainties when combining data. :contentReference[oaicite:10]{index=10}

Probability Basics

• Axioms: non-negativity, total probability $=1$, and additivity for mutually exclusive events.
• Independence means $P(A\cap B)=P(A)P(B)$; misunderstanding independence leads to the gambler’s fallacy. :contentReference[oaicite:11]{index=11}

Bernoulli’s Law of Large Numbers

• In $n$ independent, identical trials, the relative frequency ${\hat{p}}_n$ of an event converges to its true probability $p$ as $n\to \infty$. ${\hat{p}}_n\stackrel{n\to \infty }{\to }p$ :contentReference[oaicite:12]{index=12}
  • Demonstrates why long-run averages stabilise and validates frequentist probability.

Probability Mass & Density Functions

• Probability Mass Function (PMF) $p_X(x)=P(X=x)$ for discrete $X$.
• Probability Density Function (PDF) $f_X(x)$ satisfies $P(a\le X\le b)={\int }_a^b{f}_X(x)dx$ for continuous $X$. :contentReference[oaicite:13]{index=13}

Cumulative Distribution Function

• CDF $F_X(x)=P(X\le x)$ is non-decreasing, right-continuous, and satisfies $F_X(-\infty )=0$, $F_X(\infty )=1$. :contentReference[oaicite:14]{index=14}
  • For a discrete variable, $F_X$ jumps at each possible value; for a continuous variable, $F_X$ is smooth, and $F_X^{\prime }(x)=f_X(x)$. :contentReference[oaicite:15]{index=15}

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Formula Sheet & When to Use

1. Relative-frequency estimate of $p$: \hat p_n=\dfrac{\text{# successes}}{n}
   • Apply in Monte Carlo or large-sample experiments (Bernoulli’s law).
2. PMF (discrete): $p_X(x)=P(X=x)$
   • Use to list probabilities for each outcome of finite/countable $X$.
3. PDF (continuous): $f_X(x)\ge 0$, ${\int }_{-\infty }^{\infty }{f}_X(x)dx=1$
   • Integrate to find probabilities on intervals.
4. CDF (general): $F_X(x)=P(X\le x)$
   • Differentiate to get $f_X(x)$ (if continuous) or difference to get $p_X(x)$ (if discrete).
5. Bernoulli law bound: $|{\hat{p}}_n-p|<\epsilon$ with high confidence for sufficiently large $n$.

Tip: Use box plots and histograms to visualise empirical distributions before fitting theoretical models.

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