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AP Statistics 4.7: Random Variables and Probability Distributions

Intro to Random Variables and Probability Distributions

Random variables:: numerical outcomes of random behavior

  • ex:
    • X = the number of children in a randomly selected household
    • W = the time (minutes) it takes a randomly selected person to run a mile
  • In each case, the outcome of an individual event is unknown

Discrete random variable:: a random variable that can only take a countable number of values

  • ex:
    • X = the number of children in a randomly selected household
    • Possible values of X include 1, 2, 3, 4, etc…
    • It wouldn’t be logical to have 1.2 or 4.5 children
  • *countable can mean infinite, but if all the possible values were on a number line, there would be spaces between each value

Continuous random variable:: a random variable that can take an infinite number of values in an interval on a number line

  • ex:
    • W = the time (minutes) it takes a randomly selected person to run a mile
    • Possible values of W include 3.43, 4, 5, 6.8, 7, etc…
  • if all the possible values were on a number line, there would be no spaces between each value

Probability distribution:: a display of the entire set of values with their associated probability

  • the sum of all probabilities must be zero and each probability must be between 0 and 1, inclusive

Example of a probability distribution:

X1234567
P(X)3/2315/234/230001/23
  • X = the number of children in a randomly selected household
  • P(X) = the probability that a randomly selected household from this sample has X children
  • There are 23 households in this sample

Refer to the probability distribution above. Describe each probability in words and solve.

  1. P(1)

    1. the probability that a randomly selected household has 1 child
    2. P(1) = 3/23

  2. P(2)

    1. the probability that a randomly selected household has 2 children
    2. P(2) = 15/23

  3. P(X < 4)

    1. the probability that a randomly selected household has less than 4 children
    2. P(X < 4) = P(3) + P(2) + P(1) = 22/23

Interpreting a Probability Distribution

Use SOCS to describe a probability distribution like any other distribution.

  • Shape

    • How many peaks are there?
    • Is it roughly symmetric or skewed?
    • On a histogram with X on the x-axis and frequency on the y-axis, the example distribution is unimodal and roughly symmetric

  • Outliers

    • In the example, X=7 is a potential outlier

  • Center

    • Mean (in probability distributions, expected value) commonly used

  • Spread

    • Standard deviation commonly used