Choosing a distribution: 7 exam questions solved step by step

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What this lesson covers

  • Each distribution has conditions, and the conditions decide the model, not the number it produces.
  • Name the distribution and its parameters every time, or the answer cannot earn a grade.
Key idea

7 questions, one routine

  • 1. Define X in words, with units.
  • 2. Name the model and its parameters.
  • 3. State the conditions in context.
  • 4. Calculate, round, say it.
Your turn

Question 1 · Achieved, in context

  • Each kōwhai seed germinates with
  • probability 0.85. 12 seeds are sown.
  • Find P(exactly 10 germinate).
Worked example

Q1: binomial

  • Count out of a fixed n.
  • Conditions, then calculate.
  1. X = seeds that germinate of 12
  2. X ~ Bin(12, 0.85)binomial
  3. fixed n: 12 seedsgerminate or not
  4. p = 0.85 for every seedconstant p
  5. seeds germinate independently
  6. P(X = 10) = 0.292358binomial
  7. = 0.29244 d.p.
  8. about a 29% chance
Your turn

Question 2 · Achieved

  • Bags of flour labelled 1 kg have masses
  • with mean 1.02 kg and sd 0.015 kg,
  • bell-shaped about the mean. What
  • proportion of bags are under 1 kg?
Worked example

Q2: normal

  • A measurement: continuous.
  • Symmetric: normal.
  1. X = mass of a bag, in kg
  2. X ~ N(μ = 1.02, σ = 0.015)bell-shaped, continuous
  3. P(X < 1)
  4. z = (1 − 1.02) ÷ 0.015
  5. = −1.3333below the mean
  6. P(X < 1) = 0.091211
  7. = 0.09124 d.p.
  8. about 9% of bags are under 1 kg
Your turn

Question 3 · Merit

  • Flaws in a roll of fabric occur at an
  • average rate of 1.5 per metre.
  • Find P(at least one flaw in a 2 m length).
  • Careful: per metre, but 2 m.
Worked example

Q3: Poisson

  • Count in an interval.
  • Scale λ to that interval.
  1. X = flaws in a 2 m length
  2. flaws occur independently
  3. at a constant average rate
  4. X ~ Po(1.5)that is for 1 m
  5. λ = 1.5 × 2 = 3rate for 2 m
  6. X ~ Po(3)
  7. P(X ≥ 1) = 1 − P(X = 0)
  8. = 1 − 0.049787
  9. = 0.950213 ≈ 0.95024 d.p.
Your turn

Question 4 · Merit

  • A 2 m rope is equally likely to break
  • at any point along its length. Find
  • P(the shorter piece is under 0.5 m).
Worked example

Q4: uniform

  • Equally likely: flat.
  • Either end can be short.
  1. X = break point, m from one end
  2. X uniform from 0 to 2equally likely
  3. height = 1 ÷ 2 = 0.5total area 1
  4. shorter < 0.5: X < 0.5 or X > 1.5
  5. P(X < 0.5) = 0.5 × 0.5 = 0.25
  6. P(X > 1.5) = 0.5 × 0.5 = 0.25
  7. total = 0.25 + 0.25 = 0.5
  8. a 50% chance
Your turn

Question 5 · Merit, in context

  • A drive to Auckland Airport takes at
  • least 25 min, most often 35 min, and
  • at most 60 min. Choose a model and
  • find P(the drive takes over 50 min).
Worked example

Q5: triangular

  • Min, mode, max only.
  • The tail is a triangle.
  1. X = drive time, in minutes
  2. triangular: 25, 35, 60min, mode, max
  3. long tail to 60: skewedso not normal
  4. peak height = 2 ÷ 35 = 2/35
  5. height at 50 = 10/25 × 2/35similar triangles
  6. = 4/175
  7. P(X > 50) = ½ × 10 × 4/175triangle area
  8. = 4/35 = 0.11434 d.p.
  9. about 11% of drives over 50 min
Your turn

Question 6 · Excellence

  • A region feels an average of 2
  • earthquakes a month. Find P(5 or more
  • in a month). Last month there were 9,
  • after one large quake. Judge the model.
Worked example

Q6: the Poisson answer

  • A count per month.
  • Use the complement.
  1. X = quakes felt in a month
  2. X ~ Po(2)rate 2 per month
  3. P(X ≥ 5) = 1 − P(X ≤ 4)
  4. = 1 − 0.947347
  5. = 0.052653 ≈ 0.05274 d.p.
Worked example

Q6: do the conditions hold?

  • Check each condition.
  • In this context.
  1. P(X ≥ 9) = 0.000237if Po(2) holds
  2. aftershocks follow a big quake
  3. independentquakes trigger quakes
  4. constant raterate jumps after one
  5. Po(2) did not fit last month
  6. Po(2) suits quiet months only
Your turn

Question 7 · Excellence

  • A box holds 20 phones, 4 faulty.
  • 5 are picked without replacement.
  • Malia uses Bin(5, 0.2) for the faulty
  • count. Judge it; find P(none faulty).
Worked example

Q7: is it binomial?

  • Check every condition.
  • Does p stay the same?
  1. X = faulty phones of the 5
  2. fixed n = 5, faulty or not
  3. 1st pick: p = 4/20 = 0.2
  4. 2nd pick: 4/19 or 3/19no replacement
  5. p is not constant
  6. picks are not independent
  7. X ~ Bin(5, 0.2)conditions fail
Worked example

Q7: the right probability

  • Multiply pick by pick.
  • Compare with binomial.
  1. P(X = 0) = 16/20 × 15/19 × 14/18
  2. × 13/17 × 12/16
  3. = 91/323 = 0.28174 d.p.
  4. Bin: 0.8⁵ = 0.3277too high
  5. box of 2000: 0.3273p barely changes
  6. then Bin(5, 0.2) is fair
  7. for this box: P(X = 0) = 0.2817
Summary

The method that solved all seven

  • Define X, name the model, state the conditions in context, then calculate.
StepHow
Define Xin words, with units
Name itmodel and parameters
Conditionsin the context
Poissonscale λ to the interval
Calculatestatement, result, round
Judgewhich condition could fail

More in AS 91586

Choosing a distribution, and naming it every time7 minDiscrete random variables: mean and standard deviation6 minBinomial and Poisson: at least, at most, and the rate7 minThe normal distribution: forwards, backwards and conditional7 minUniform and triangular distributions, and judging a model7 minChoosing a distribution: 7 exam questions solved step by step10 minDiscrete random variables: 7 exam questions solved step by step11 minBinomial and Poisson: 7 exam questions solved step by step10 minNormal distribution: 7 exam questions solved step by step10 minUniform and triangular: 7 exam questions solved step by step10 min

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