Discrete Random Variables Section 6.1

Objectives Distinguish between discrete and continuous random variables Identify discrete probability distributions Construct probability histograms Compute and interpret the mean of a discrete random variable Interpret the mean of a discrete random variable as an expected value Compute the variance and standard deviation of a discrete random variable

Distinguish Between Discrete and Continuous Random Variables Def: Random variable A numerical measure of the outcome of a probability experiment. Its value is determined by chance Denoted using letters, such as X. Example: Coin Flip Experiment X represents the number of heads in two flips of a coin. Possible values of X: 0, 1, 2

More Definitions Discrete Random Variable Has either a finite or countable number of values Values can be plotted on a number line with space between each point Continuous Random Variable Infinitely many values Can be plotted on a line in an uninterrupted fashion

Discrete or Continuous? Speed of a space shuttle Acceleration of a car Number of cars at an intersection in 15 minute segments Number of heart beats in 1 minute segment

Identify Discrete Probability Distributions Def: Probability Distribution Of a discrete random variable, X, provides the possible values of the random variable and their corresponding probabilities Can be described via a table, graph or formula Rules for a Discrete Probability Distribution Sum of the probabilities in the distribution equals 1 Each probability in the distribution will be: 0 < P(x) < 1

Construct Probability Histogram Def: Probability Histogram Histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable PROBABILITYPROBABILITY Values of Random Variable

Compute the Mean of a Discrete Random Variable We can describe the distribution of a variable Center, spread, shape (mean/median, SD/Var, Skew) Now we will examine methods to identify the center and spread of a discrete random variable Mean, SD/Variance

Compute the Mean of a Discrete Random Variable Mean of a discrete random variable: μ x =Σ [x * P(x)] where x is the value of the random variable and P(x) is the probability of observing the random variable x.

Example Probability Distribution for the number of students absent from a statistics class. Compute the mean: Students, x Probability, P(x) Total1.00

Interpretation of the Mean of a Discrete Random Variable So, what does the mean tell us? It is the average outcome if the experiment is repeated MANY, MANY times. Remember, μ x, refers to a population mean. X-bar refers to a sample mean. The more times the experiment is repeated, the closer x- bar gets to μ x.

Interpretation of the Mean of a Discrete Random Variable as an Expected Value Because μ x represents what happens in the long-run, we can also call it the “expected value”. Therefore, when you hear someone refer to the expected value or the interpretation of the expected value, they are referring to the mean of the discrete random variable.

Compute the Variance and SD of a Discrete Random Variable The variance of a discrete random variable, σ 2 x, is a weighted average of the squared deviations where the weights are the probabilities.

Example Probability Distribution for the number of students absent from a statistics class. Compute the variance and standard deviation: Students, x Probability, P(x) Total1.00

Assignment Pg : 1, 2, 3, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23