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AI SL · 13.2

Probability Distributions

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13.2.10

Distributions review: chapter 8 checklist and short quiz

13.2.10

Distributions review: chapter 8 checklist and short quiz

Target: I can find an unknown probability in a table, then find E(X) and a probability for a discrete variable. (+4 more)

By the end of this section

Key points

Discrete distributions

  • In any discrete distribution the probabilities add to 1. Use this to find an unknown k.
  • \(E(X) = \sum x\,P(X = x)\): multiply each value by its probability and add.
  • 'At least 2' means 2 or more: add those probabilities, or use 1 − P(X < 2).

The binomial distribution

  • \(Y \sim B(n, p)\): a fixed number n of independent trials, two outcomes, constant probability p.
  • \(E(Y) = np\) and \(\text{Var}(Y) = np(1 - p)\).
  • Exactly k: binomial pdf. At most k: binomial cdf. At least k: 1 − P(Y ≤ k − 1).

The normal distribution

  • Normal cdf takes a lower and upper limit. For a tail, use 10^99 or −10^99 as the missing limit.
  • Inverse normal takes the area to the left. For an upper tail of p, use the area 1 − p.
  • Expected number = probability × sample size.
  • Which function? A count of successes: binomial. A measurement: normal.

On the calculator

Binomial pdf and cdf

  • Enter n, p and the value x.
  • pdf gives P(Y = x). cdf gives P(Y ≤ x).

Screenshot and video to come

Normal cdf

  • Enter lower, upper, \(\mu\) and \(\sigma\).
  • Use 10^99 or −10^99 for a tail.

Screenshot and video to come

Inverse normal

  • Enter the area to the left, \(\mu\) and \(\sigma\).
  • For an upper tail of p, use the area 1 − p.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A discrete distribution

The random variable X has the probability distribution shown. Find k, then find E(X) and P(X ≥ 2).

x1234P(X = x)0.12k0.282k
Working
  1. Sum to 1

    0.12 + k + 0.28 + 2k = 1

    In any discrete distribution the probabilities add to 1.

  2. Solve for k

    0.4 + 3k = 1, so k = 0.2

    The probabilities are 0.12, 0.2, 0.28, 0.4.

  3. Expected value

    \(E(X) = 1 \times 0.12 + 2 \times 0.2 + 3 \times 0.28 + 4 \times 0.4 = 2.96\)

    Multiply each value by its probability and add.

  4. At least 2

    P(X ≥ 2) = 0.2 + 0.28 + 0.4 = 0.88

    Or 1 − P(X = 1) = 1 − 0.12.

Video walkthrough coming
A binomial in context

A quiz has 20 multiple-choice questions, each with 4 options. A student guesses every answer. Let Y be the number of correct answers. Write down E(Y), and find the probability that the student gets at least 8 correct.

Working
  1. The distribution

    \(Y \sim B(20, 0.25)\)

    Fixed number of questions, two outcomes each, independent, constant p = 0.25.

  2. Expected value

    \(E(Y) = np = 20 \times 0.25 = 5\)

    Write down E(Y) = np. No calculator is needed.

  3. Complement

    \(P(Y \geq 8) = 1 - P(Y \leq 7)\)

    A cdf gives 'at most'. Turn 'at least 8' into 1 − P(Y ≤ 7).

  4. Calculator

    From the calculator: binomial cdf with n = 20, p = 0.25, upper = 7: 0.8982

    Upper = 7 gives P(Y ≤ 7).

  5. Answer

    \(P(Y \geq 8) = 1 - 0.8982 = 0.102\)

    Give 3 significant figures.

Video walkthrough coming
Normal: all three skills

The mass of a plum is normally distributed with mean 45 g and standard deviation 6 g. (a) Find the probability that a plum has mass between 40 g and 55 g. (b) The heaviest 12% of plums are sold as large. Find the least mass of a large plum. (c) In a sample of 500 plums, find the expected number with mass less than 38 g.

Working
  1. (a) Between

    From the calculator: normal cdf with lower = 40, upper = 55, μ = 45, σ = 6: 0.750

    A probability between two values uses lower and upper.

  2. (b) Inverse normal

    From the calculator: inverse normal with area = 0.88, μ = 45, σ = 6: 52.0

    Heaviest 12% is an upper tail, so the area to the left is 0.88.

  3. (c) Probability

    From the calculator: normal cdf with lower = −10^99, upper = 38, μ = 45, σ = 6: 0.1217

    First the probability for one plum.

  4. (c) Expected number

    \(500 \times 0.1217 = 60.8\)

    About 61 plums. Expected number = probability × sample size.

Video walkthrough coming