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:

• 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