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AI HL · 13.2

Probability Distributions

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13.2.11

Poisson probabilities, mean and variance, and rescaling the rate

13.2.11

Poisson probabilities, mean and variance, and rescaling the rate

Target: I can find P(X = k) for a Poisson variable with the calculator's Poisson pdf. (+3 more)

By the end of this section

Key points

pdf or cdf?

  • Write X ~ Po(λ) first. λ must be the average number of events in the period X counts.
  • Poisson pdf gives one value: \(P(X = k)\).
  • Poisson cdf gives a running total: \(P(X \le k)\), adding from 0 up to k.
  • Write the settings you used (λ and x, or λ, lower and upper) as part of your working.

Turning words into cdf calls

  • At most k: \(P(X \le k)\). Fewer than k: \(P(X \le k-1)\).
  • At least k: \(1 - P(X \le k-1)\). More than k: \(1 - P(X \le k)\).
  • X has no largest value, so "at least" always uses the complement.
  • Between a and b, inclusive: \(P(X \le b) - P(X \le a-1)\).

Mean, variance and standard deviation

  • If \(X \sim \text{Po}(\lambda)\) then \(E(X) = \lambda\) and \(\text{Var}(X) = \lambda\).
  • The standard deviation is \(\sqrt{\lambda}\). Do not give the variance when the question asks for the standard deviation.
  • A mean equal to the variance is a clue that data fit a Poisson model. It is not proof, because other distributions have it too.

Changing the period

  • The mean scales with the length of the period. 15 per hour is 5 per 20 minutes, and 30 per 2 hours.
  • Rewrite λ for the period in the question before you use the calculator: \(\lambda_{\text{new}} = \text{rate} \times \text{length of period}\).
  • Then write the new distribution, for example \(X \sim \text{Po}(5)\), and find the probability as usual.

On the calculator

Poisson pdf: one value

  • Enter λ and x.
  • Write down the settings: λ, x.

Screenshot and video to come

Poisson cdf: a running total

  • Enter λ, lower and upper.
  • For P(X ≤ k) use lower = 0 and upper = k.
  • For at least k, find P(X ≤ k−1) and take it from 1.

Screenshot and video to come

Explore: the Poisson distribution

X counts the events in a period, at a constant average rate. Set the rate and the number of periods, then choose which values of X to add up, or press a bar.

=

λ = m × t
E(X) = λ
Var(X) = λ
Standard deviation, √λ

Move a slider to change the distribution.

  1. Set m = 4, then change t from 1 to 2 to 3. What does t do to the mean? What happens to the shape?
  2. Compare the mean with the variance each time you move a slider. What do you notice? Is that true for a binomial distribution?
  3. Move the upper slider to its largest value (∞). Which probability do you take away from 1 to get the same answer?
  4. Add the second variable. Where does the gold curve for X + Y sit compared with the two means? Why must the means add?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Exactly k

A seismologist records tremors in a region at a constant average rate of 4.5 per month, independently of each other. X is the number of tremors next month. Find the probability that exactly 3 tremors occur.

Working
  1. Define X

    Let X be the number of tremors next month. X ~ Po(4.5)

    Say what X counts, then write the distribution.

  2. Poisson pdf

    From the calculator: Poisson pdf with λ = 4.5, x = 3

    The pdf gives one value of X.

  3. Answer

    P(X = 3) = 0.169

    3 significant figures.

Video walkthrough coming
At least k

Hawks are seen over a ridge independently of each other, at a constant average rate of 2.5 per hour. X is the number of hawks seen in the next hour. Find the probability that at least 4 hawks are seen.

Working
  1. Define X

    X ~ Po(2.5)

    State the distribution.

  2. Use the complement

    P(X ≥ 4) = 1 − P(X ≤ 3)

    Everything except 0, 1, 2, 3. There is no largest value of X, so a cdf cannot stop at an upper limit.

  3. Poisson cdf

    From the calculator: Poisson cdf with λ = 2.5, lower = 0, upper = 3

    Lower 0, upper 3.

  4. Subtract

    P(X ≤ 3) = 0.7576, so P(X ≥ 4) = 1 − 0.7576 = 0.242

    3 significant figures at the end.

Video walkthrough coming
Mean and standard deviation

An insect trap catches flies at a constant average rate of 14 per day, independently of each other. X is the number of flies caught in a day. Write down the mean of X and find the standard deviation of X.

Working
  1. Define X

    X ~ Po(14)

    λ = 14 per day.

  2. The mean

    E(X) = λ = 14

    The mean of a Poisson variable is λ.

  3. The variance

    Var(X) = λ = 14

    The variance is also λ.

  4. Standard deviation

    standard deviation = √14 = 3.74

    The standard deviation is the square root of the variance.

Video walkthrough coming
A shorter period

A bakery sells loaves independently of each other, at a constant average rate of 21 per hour. Find the probability that fewer than 5 loaves are sold in a 15 minute period.

Working
  1. Rescale the mean

    15 minutes is 1/4 of an hour, so λ = 21 ÷ 4 = 5.25

    The mean scales with the length of the period.

  2. Define X

    Let X be the number of loaves sold in 15 minutes. X ~ Po(5.25)

    Write the new distribution.

  3. Whole numbers

    P(X < 5) = P(X ≤ 4)

    Fewer than 5 means 0, 1, 2, 3 or 4. The calculator gives probabilities of the form P(X ≤ k).

  4. Poisson cdf

    From the calculator: Poisson cdf with λ = 5.25, lower = 0, upper = 4

  5. Answer

    P(X ≤ 4) = 0.398

Video walkthrough coming