Statistics 3 - Geometric and negative binomial distributions

Question 1

What is the geometric distribution?

Answer 1

For successive independent trials, each with a constant probability of success, 𝑝, the number of trials needed to get one success, 𝑋, has the geometric distribution (written as 𝑋∼Geo(𝑝)), with the probability function:
P(𝑋=𝑥)=p(𝑥)=𝑝(1-𝑝)ˣ⁻¹ Where 𝑥=1,2,3,…

Question 2

How is the cumulative geometric distribution calculated?

Answer 2

The values of p(𝑥) form a geometric distribution with first term 𝑝 and common ratio, (1-𝑝). As such the cumulative geometric distribution can be derived by considering the sum of the terms of the geometric series.
ₓ
Σp(r)=𝑝(1-(1-𝑝)ˣ)/1-(1-𝑝)ˣ=𝑝(1-(1-𝑝)ˣ)/𝑝=1-(1-𝑝)ˣ
ʳ⁼¹
Thus if 𝑋∼Geo(𝑝), then the cumulative geometric distribution is given by:
P(𝑋≤x)=1-(1-𝑝)ˣ Where 𝑥=1,2,3,…

Question 3

How is the mean of a geometric distribution calculated?

Answer 3

If 𝑋∼Geo(𝑝), then:

The mean of 𝑋=E(𝑋)=μ=1/𝑝

Question 4

How is the variance of a geometric distribution calculated?

Answer 4

If 𝑋∼Geo(𝑝), then:

The variance of 𝑋=Var(𝑋)=σ²=1-𝑝/𝑝²

Question 5

What is the negative binomial distribution?

Answer 5

The binomial distribution, 𝑋∼B(𝑛,𝑝), models the number of successes in a fixed number of trials, 𝑛. The probability of success in each trial, 𝑝, is constant and the trials are independent.
The negative binomial distribution, often written as 𝑋∼NB(𝑟,𝑝) or 𝑋∼Negative B(𝑟,𝑝), models the number of trials needed to achieve a fixed number of successes, 𝑟.
For successive trials, each with the constant probability of success, 𝑝, the number of trials needed to get 𝑟 successes, 𝑋, has the negative binomial distribution with probability function:
P(𝑋=𝑥)=p(𝑥)=(ˣ⁻¹ᵣ₋₁)𝑝ʳ(1-𝑝)ˣ⁻ʳ 𝑥=𝑟,𝑟+1,𝑟+2,…
This is the probability of 𝑟-1 successes in 𝑥-1 trials multiplied by the probability of success in the 𝑥th trial

Question 6

How is the mean of a negative binomial distribution calculated?

Answer 6

If 𝑋∼NB(𝑟,𝑝), then:

The mean of 𝑋=E(𝑋)=μ=𝑟/𝑝

Question 7

How is the variance of a negative binomial distribution calculated?

Answer 7

If 𝑋∼NB(𝑟,𝑝), then:

The variance of 𝑋=Var(𝑋)=σ²=𝑟(1-𝑝)/𝑝²