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}