# Geometric and Negative Binomial Distributions

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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