Cambridge International AS & A Level Mathematics: Probability & Statistics 2 Comprehensive Study Notes

Introduction to Mathematical Problem Solving and Modelling

• Doing mathematics is likened to art: mastering tools (algebra, calculus) and theoretical ideas is the technical side, but the joy comes through creative problem-solving.
• Mathematical 'Problem' Definition: A question that a mathematician does not immediately know how to answer (non-routine). If the answer is immediate, it is an 'exercise'.
• Mathematical Modelling: Equations used to capture key aspects of real-world situations to make predictions. Examples include weather prediction, climate change, forensic science, and financial markets.
• Communication: Mathematics involves discussing ideas with colleagues; 'proof' is the heart of the skill of convincing others of correctness through verbal and written reasoning.

Chapter 1: Hypothesis Testing

• Hypothesis: A claim believed or suspected to be true.
• Hypothesis Test: A scientific analysis using sample data to determine if a claim could happen by chance or if results are statistically significant.
• Null Hypothesis ($H_0$): The default assumption that there is no difference between the usual outcome and what is being tested.
• Alternative Hypothesis ($H_1$): The hypothesis accepted if the null hypothesis is rejected ($p < k, p > k, \text{ or } p \neq k$).
• Significance Level ($\alpha$): The probability of rejecting a true null hypothesis (often 5%, 10%, or 1%). It determines the threshold for significance.
• Test Statistic: The probability or region calculated using sample data (e.g., $P(X \leq k)$) compared against the significance level.
• Critical Region (Rejection Region): The range of values for which the null hypothesis is rejected.
• Critical Value: The specific value at which the decision changes from accepting to rejecting the null hypothesis.
• Dice Experiment Example: Rolling a die 16 times. If $X \sim B(16, \frac{1}{6})$, probabilities are:
  • $P(X \geq 5) = 0.1134 \, (11.3\%)$
  • $P(X \geq 6) = 0.0378 \, (3.78\%)$
  • If the significance level is 5%, rolling 6 sixes is the critical value for bitas.
• Falsification Theory: Karl Popper's theory suggests a claim is accepted until evidence (like the sun not rising) falsifies it.
• Type I Error: Rejecting $H_0$ when it is actually true. The probability of this error is equal to the significance level.
• Type II Error: Accepting $H_0$ when it is actually false.
• Conditions for Approximation: A binomial distribution $B(n, p)$ can be approximated by a normal distribution $N(np, npq)$ for large samples, requiring a continuity correction of $\pm 0.5$.

Chapter 2: The Poisson Distribution

• Poisson Distribution ($X \sim Po(\lambda)$): Models discrete events occurring independently, singly, and at random in a fixed interval of time or space at a constant average rate.
• Prussian Army Example: Ladislaus Bortkiewicz (1898) used Poisson to model soldiers killed by horse kicks over 20 years for 14 corps (mean deaths = 0.7).
• Properties:
  • Mean $E(X) = \lambda$
  • Variance $Var(X) = \lambda$
  • Probability Formula: $P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}$, for $r = 0, 1, 2, \dots$
• Modelling: If mean and variance are approximately equal, Poisson is a suitable model.
• Adapting Intervals: The rate is proportional to the interval. If $\lambda$ is the rate for time $t$, the rate for time $kt$ is $k\lambda$.
• Poisson as Approximation to Binomial: Suitable when $n > 50$ and $np < 5$ (rare events, large trials). The variance of the binomial $np(1-p)$ approaches the mean $np$ as $p$ decreases.
• Normal Approximation to Poisson: For $\lambda > 15$, $Po(\lambda) \approx N(\lambda, \lambda)$. A continuity correction of $\pm 0.5$ is required.

Chapter 3: Linear Combinations of Random Variables

• Expectation of Constants: $E(aX + b) = aE(X) + b$.
• Variance of Constants: $Var(aX + b) = a^2 Var(X)$. Note: $Var(b) = 0$.
• Sums and Differences of Independent Variables:
  • $E(aX + bY) = aE(X) + bE(Y)$.
  • $Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$.
  • $Var(X - Y) = Var(X) + Var(Y)$ (variances always add for independent variables).
• Multiples vs. Observations:
  • $2X$: Twice the size of one observation ($Var = 4 Var(X)$).
  • $X_1 + X_2$: The sum of two independent observations ($Var = 2 Var(X)$).
• Normal Distributions: If $X$ and $Y$ are independent and normal, any linear combination $aX + bY$ is also normal.
• Poisson Distributions: If $X \sim Po(\lambda)$ and $Y \sim Po(\mu)$, then $X + Y \sim Po(\lambda + \mu)$. Multiples (e.g., $2X$) are not Poisson as mean and variance will no longer be equal ($E = 2\lambda, Var = 4\lambda$).

Chapter 4: Continuous Random Variables

• Probability Density Function (PDF): A function $f(x)$ such that:
  • $f(x) \geq 0$ for all $x$.
  • Total area under the curve is 1: $\int_{-\infty}^{\infty} f(x)\,dx = 1$.
• Probability: $P(a < X < b) = \int_{a}^{b} f(x)\,dx$. Each individual point has zero probability: $P(X = a) = 0$.
• Median ($m$): The value where the area up to that point is 0.5: $\int_{-\infty}^{m} f(x)\,dx = 0.5$.
• Expectation and Variance:
  • Mean $E(X) = \int_{-\infty}^{\infty} x f(x)\,dx$.
  • $E(X^2) = \int_{-\infty}^{\infty} x^2 f(x)\,dx$.
  • Variance $Var(X) = E(X^2) - [E(X)]^2$.
• Exponential Distribution: A specific continuous PDF $f(t) = \lambda e^{-\lambda t}$ for $t \geq 0$, often used for waiting times in Poisson processes.

Chapter 5: Sampling and the Central Limit Theorem

• Terminology:
  • Population: The complete set of items of interest (e.g., all fish in the sea).
  • Sample: A part of the population used to infer characteristics of the whole.
  • Census: Data collected from every member of the population (expensive, time-consuming).
• Central Limit Theorem (CLT): For a large sample size $n$ ($n > 50$) from any distribution with mean $\mu$ and variance $\sigma^2$, the distribution of the sample mean $\bar{X}$ is approximately normal: $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$.
• Standard Error: The standard deviation of the sample mean distribution, defined as $\frac{\sigma}{\sqrt{n}}$.
• Continuity Correction for Sample Means: When sampling from discrete distributions, the correction is $\pm \frac{1}{2n}$.

Chapter 6: Estimation

• Point Estimates:
  • Unbiased Estimate of Population Mean: $\bar{x} = \frac{\sum x}{n}$.
  • Unbiased Estimate of Population Variance ($s^2$): $\frac{1}{n-1} (\sum x^2 - n\bar{x}^2)$. The division by $n-1$ (Bessel's correction) corrects the bias of the sample variance.
• Confidence Intervals (CI): A range of values likely to contain the true population parameter.
• CI for Population Mean (Known $\sigma^2$): $\bar{x} \pm z \frac{\sigma}{\sqrt{n}}$. Common critical values ($z$):
  • 90% CI: $z = 1.645$
  • 95% CI: $z = 1.960$
  • 99% CI: $z = 2.576$
• CI for Sample Proportion ($p$): For large $n$, the interval is $\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. Width is determined by the size of $n$ and confidence level.