Introduction

• A random variable takes numerical values that describe the outcomes of a chance process

  • Use capital letters to designate random variables

• The probability distribution of a random variable gives it possible values and their probabilities

  • Two main types: discrete and continuous

Discrete Random Variables

• A discrete random variable, X, takes a fixed set of possible values with gaps between them

• Probability Distribution for a Discrete Random Variable

  • The probability distribution of a discrete random variable, X, list the values x, and their probabilities p:

• For the probability distribution to be valid, the probabilities must satisfy two requirements:

  • every probability, p, is a number between 1 and 0

  • the sum of the probabilities is 1

Analyzing Discrete Random Variables: Describing Change

• Display probability distribution with a histogram

  • Values of variable go on the horizontal axis

  • Probability goes on the vertical axis

• A probability distribution histogram is really just a relative frequency histogram because probabilities are long-run frequencies.

Measuring Center: The Mean (Expected Value) of a Discrete Random Variable

• The mean x̄ of a quantitative data set with n observations is

  • x̄ = sum of data values/number of data values

• The mean of any discrete random variable is an average of the possible outcomes, but a weighted average in which each outcome is weighted by its probability

• The mean (expected value) of a discrete random variable is its average value over many, many repetitions of the same chance process.

  • Suppose X is a discrete random variable with probability distribution:

• To find the mean (expected value) of X, multiply each possible value of X by its probability then add all of the products.

Measuring Variability: The Standard Deviation (and Variance) of a Discrete Random Variable

• The standard deviation of a discrete random variable measures how much the values of the variable typically vary from the mean.

  • Suppose that X is a discrete random variable with the probability distribution

  • and that μ⌄x is the mean of X. The variance of X is

    • Var(X)=σ^2⌄x = (X⌄1 - μ⌄x)^2* p⌄1 + (X⌄2 - μ⌄x)^2* p⌄2 + (X⌄3 - μ⌄x)^2* p⌄3 + …

  • The standard of deviation of X is the square root of the variance.

• Using a Calculator to Analyze Discrete Random Variables: TI-83/84

  • Enter the values of the random variable list in L1 and the corresponding probabilities in list L2

  • To graph a histogram of the probability distribution:

    • In the stat plot menu define Plot 1 to be a histogram with Xlist: L1 and Freq: L2

    • Adjust your window settings as follows: Xmin = -1, Xmax = 11, Xscl = 1, Ymin = -0.1, Ymax = 0.5, Yscl = 0.1.

    • Press GRAPH

  • To calculate the mean and standard deviation of the random variable, use one-variable statistics with the values in L1 and the probabilities (relative frequencies) in L2. Press STAT, arrow to the CALC menu, and choose 1-Var Stats.

    • OS 2.55 or later: In the dialog box, specify List: L1 and FreqList: L2. Then choose Calculate.

    • Older OS: Execute the command 1-Var Stats L1,L2.

Continuous Random Variables

• A continuous random variable can take any value in an interval on the number line

• How to Find Probabilities for a Continuous Random Variable

  • The probability of any event involving a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that make up the event

• The probability for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes

Summary

• A random variable takes numerical values determined by the outcome of a chance process. The probability distribution of a random variable gives its possible values and their probabilities. There are two types of random variables: discrete and continuous.

• A discrete random variable has a fixed set of possible values with gaps between them.

  • A valid probability distribution assigns each of these values a probability between 0 and 1 such that the sum of all the probabilities is exactly 1.

  • The probability of any event is the sum of the probabilities of all the values that make up the event.

  • We can display the probability distribution as a histogram, with the values of the variable on the horizontal axis and the probabilities on the vertical axis.

• A continuous random variable can take any value in an interval on the number line.

  • A valid probability distribution for a continuous random variable is described by a density curve with area 1 underneath.

  • The probability of any event is the area under the density curve directly above the values on the horizontal axis that make up the event.

• We can describe the shape of a probability distribution histogram or density curve in the same way as we did a distribution of quantitative data—by identifying symmetry or skewness and any major peaks.

• Use the mean to summarize the center of a probability distribution. The mean of a random variable μX is the balance point of the probability distribution histogram or density curve.

  • The mean is the long-run average value of the variable after many repetitions of the chance process. It is also known as the expected value of the random variable, E(X).

  • If X is a discrete random variable, the mean is the average of the values of X, each weighted by its probability:

    • μX=E(X)= ∑ x⌄i * p⌄i = x⌄1 p⌄1 + x⌄2 * p⌄2 + x⌄3 * p⌄3 + ···

• Use the standard deviation to summarize the variability of a probability distribution. The standard deviation of a random variable σX measures how much the values of the variable typically vary from the mean.

  • If X is a discrete random variable, the variance of X is the “average” squared deviation of the values of the variable from their mean:

    • σX2=∑ (x⌄i−μ⌄X)^2*p⌄i = (x⌄1−μ⌄X)^2 * p⌄1 + (x⌄2−μ⌄X)^2 p⌄2 + (xV3−μ⌄X)^2 * p⌄3 + ...

• The standard deviation of X is the square root of the variance.