Students learn how to recognise and use the geometric and negative binomial distributions throughout this unit. Later, as learning progresses, they find the mean and variance of both distributions.
Lessons on Geometric and Negative Binomial Distributions
From Statistics 1
From Further Statistics 1 AS
P(X=r)=(1-p)^{r-1} p \ \ \ \ \ x = 1, 2, 3, ...E(x)=\frac{1}{p}\operatorname{Var}(x)=\frac{1-p}{p^{2}}p(X=x)=\left(\begin{array}{c}x-1 \\ r-1\end{array}\right) p^{r}(1-p)^{x-r}\ x = r, r + 1, ...
E(x)=\frac{r}{p}\operatorname{Var}(x)=\frac{r(1-p)}{p^{2}}P(X \leq x)=1-(1-p)^{x}P(X>x)=(1-p)^{x}P(X \geqslant x)=(1-p)^{x-1}
Students learn how to apply the geometric distribution to model independent trials with a constant probability of success. They calculate probabilities, including P(X = r) and P(X > x) , using real-world examples like rolling a dice or testing defective components.
Later, as learning progresses, students derive the cumulative probability function and assess when the model applies.
Introducing Geometric Distribution
Cumulative Geometric Distribution
Students learn how to calculate the mean (E(X)) and variance (Var(X)) of a geometric distribution. They apply the probability mass function to determine specific probabilities, such as P(X = k).
Later, as learning progresses, they use expectation and variance to analyse probability distributions, interpret real-world scenarios, and justify assumptions.
Students learn to apply the negative binomial distribution to problems involving a fixed number of successes in a sequence of trials. They explore the notation X ~ NB(r, p) and use the probability mass function to calculate specific probabilities.
Later, they apply their knowledge to real-world scenarios like dice rolls and chess games, solving problems involving cumulative and conditional probabilities.
Students learn how to calculate the mean and variance of a negative binomial distribution.
Later, as learning progresses they use the mean and variance to calculate probabilities.
Students learn how to test against the probability in a geometric distribution.
Later, as learning progresses they move from one- to two-tail tests.
Students learn how to apply the geometric distribution to hypothesis testing, using probability functions to determine critical regions at a 5% significance level. They test null and alternative hypotheses, calculating actual significance levels to understand Type I errors.
Real-world contexts, such as spinner experiments and quality control, reinforce practical applications.
Students learn how to test a goodness of fit for a geometric distribution.
Later, as learning progresses they find a geometric model by working out the probability from the mean average and check it using a chi-squared test.
A research-backed case exploring why over-reliance on automated math homework platforms and repetitive worksheets lowers student expectations, and how departments can build genuine mathematical thinking.
How to teach converting between fractions, decimals and percentages.
Four problem solving lessons to develop student’s mathematical reasoning and communication skills.