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**Further A-Level Mathematics Year 2 Statistics 1: Geometric and Negative Binomial Distributions**

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**

**Prerequisite Knowledge**

From Statistics 1

- Use simple, discrete probability distributions, including the binomial distribution;
- Identify the discrete uniform distribution;
- Calculate probabilities using the binomial distribution.
- Calculate cumulative probabilities using the binomial distribution

*From Further Statistics 1 AS*

- Find the expectation E(X) of a discrete random variable X and understand its meaning
- Find the variance Var (X) of a discrete random variable X and understand its meaning
- Solve a range of problems involving discrete random variables

**Success Criteria**

- Recognise situations when the geometric distribution is likely to be a suitable model.
- Calculate probabilities using the geometric distribution formula.
- Calculate the mean and variance of a geometric distribution
- Recognise situations when the negative binomial distribution is likely to be a suitable model.
- Calculate probabilities using a negative binomial distribution
- Calculate the mean and variance of a negative binomial distribution

**Teaching Points**

- The geometric distribution may apply when:
- There are two outcomes, success and failure
- Both outcomes have fixed probabilities,
- Probabilities remain constant
- You are finding the number of trials it takes for the first success.
- The probability of success in any trial is independent of the outcome of any other trial.

- For a geometric random variable X, where X âˆ¼ Geo (p)

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}}

- The negative binomial distribution may be used in situations in which
- There are two outcomes, success and failure
- Both outcomes have fixed probabilities
- You are finding the number of trials it takes for the rth success to occur.
- Probabilities remain constant
- The probability of success in any trial is independent of the outcome of any other trial.

- For a negative binomial random variable X, with parameters p (probability) and r (number of successes)

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}}

**Misconceptions**

- Students can find it challenging to determine whether a geometric or negative binomial distribution may apply to the situation.
- Some students apply the correct distribution but not the associated formulae. Encourage students to use the formulae booklet.
- Students need to be able to derive the formulae for the geometric distribution as these are not in the formula booklet.

P(X \leq x)=1-(1-p)^{x}

P(X>x)=(1-p)^{x}

P(X \geqslant x)=(1-p)^{x-1}

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