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Step 17: Statistics and probability

Random variables: binomial and normal

Step 17 · Statistics and probability topics

Find the mean and standard deviation of a random variable, use the binomial distribution for counts of successes, and use the normal distribution and z-scores for measurements.

Expected value and spread

A random variable X takes numerical values with certain probabilities. Its expected value, or mean, is the probability-weighted average of its values: the long-run average over many repeats.

E(X)=μ=∑x P(x)σ2=∑(x−μ)2P(x)E(X) = \mu = \sum x\,P(x) \qquad \sigma^{2} = \sum (x - \mu)^{2} P(x)
x012
P(x)0.250.50.25

Example: Two coin tosses

X is the number of heads in two fair tosses, with the distribution in the table. Find its mean and standard deviation.

  1. μ = 0(0.25) + 1(0.5) + 2(0.25), which is 1.
  2. σ² = 1(0.25) + 0(0.5) + 1(0.25), which is 0.5.
  3. σ = √0.5.

AnswerMean 1, standard deviation about 0.71

Example: A game

A game costs $2. You win $10 with probability 0.15, and otherwise nothing. What is your expected gain?

  1. Expected winnings: 10 × 0.15 = 1.5.
  2. Take away the $2 cost.

Answer−$0.50 a game: a loss in the long run

The binomial distribution

Repeat a trial n times, independently, each with the same success probability p. The number of successes X is binomial. Its probabilities come from counting the ways to place k successes among n trials.

P(X=k)=(nk)pk(1−p)n−kμ=npσ=np(1−p)P(X = k) = \binom{n}{k}p^{k}(1 - p)^{n-k} \qquad \mu = np \qquad \sigma = \sqrt{np(1 - p)}

Example: Exactly two

A quiz has 5 questions, each guessed right with probability 0.25. What is the probability of exactly 2 right?

  1. There are (52)=10\binom{5}{2} = 10 ways to choose which two are right.
  2. Each way has probability 0.25² × 0.75³, about 0.0264.
  3. Ten times that.

AnswerAbout 0.264

Example: Mean and spread

A player makes 70% of free throws. In 50 throws, what are the mean and standard deviation of the number made?

  1. μ = np, which is 35.
  2. σ = √(50 × 0.7 × 0.3), which is √10.5.

AnswerMean 35, standard deviation about 3.24

The normal distribution

The normal distribution is the symmetric bell curve, fixed by its mean μ and standard deviation σ. A z-score says how many standard deviations a value is from the mean.

z=x−μσz = \frac{x - \mu}{\sigma}
A normal curve, with dashed lines 1 and 2 standard deviations from the mean. About 68% of values lie within 1 SD of the mean, 95% within 2 and 99.7% within 3.

Example: The 68-95-99.7 rule

Adult heights in a population have mean 64 inches and SD 2.5. About what share are between 59 and 69 inches?

  1. 59 and 69 are 2 SDs below and above 64.
  2. About 95% of values lie within 2 SDs.

AnswerAbout 95%

Example: A z-score

An exam has mean 70 and SD 8. What is the z-score of 82, and about what share score higher?

  1. z = (82 − 70) ÷ 8, which is 1.5.
  2. A normal table gives 0.9332 below z = 1.5.

Answerz = 1.5; about 6.7% score higher

Try it yourself

Work each one on paper first, then open the answer to check.

  1. X is 1 with probability 0.3 and 0 otherwise. What is E(X)?

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    0.3

  2. X is 0, 10 or 20 with probabilities 0.5, 0.3 and 0.2. What is E(X)?

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    7

  3. A binomial variable has n = 20 and p = 0.4. What is its mean?

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    8

  4. Flip a fair coin 4 times. What is the probability of exactly 2 heads?

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    616=38\frac{6}{16} = \frac{3}{8}

  5. Scores have mean 500 and SD 100. What is the z-score of 650?

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    1.5

  6. Using the 68-95-99.7 rule, what share of a normal distribution lies more than 1 SD above the mean?

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    About 16%

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