1. Discrete Random Variables
Chapter 5
Discrete Random Variables

2. Chapter Outline
5.1 Two Types of Random Variables 5.2 Discrete Probability Distributions 5.3 The Binomial Distribution 5.4 The Poisson Distribution (Optional) 5.5 The Hypergeometric Distribution (Optional) 5.6 Joint Distributions and the Covariance (Optional)

3. 5.1 Two Types of Random Variables
LO 5-1: Explain the difference between a discrete random variable and a continuous random variable.
5.1 Two Types of Random Variables
Random variable: a variable that assumes numerical values determined by the outcome of an experiment
Discrete
Continuous
Discrete random variable: Possible values can be counted or listed
The number of defective units in a batch of 20
A rating on a scale of 1 to 5
Continuous random variable: May assume any numerical value in one or more intervals
The waiting time for a credit card authorization
The interest rate charged on a business loan

4. 5.2 Discrete Probability Distributions
LO 5-2: Find a discrete probability distribution and compute its mean and standard deviation.
5.2 Discrete Probability Distributions
The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assume
Denote the values of the random variable by x and the value’s associated probability by p(x)

5. 5.3 The Binomial Distribution
LO 5-3: Use the binomial distribution to compute probabilities.
5.3 The Binomial Distribution
The binomial experiment…
Experiment consists of n identical trials
Each trial results in either “success” or “failure”
Probability of success, p, is constant from trial to trial
Trials are independent
If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable

6. 5.4 The Poisson Distribution (Optional)
LO 5-4: Use the Poisson distribution to compute probabilities (Optional).
5.4 The Poisson Distribution (Optional)
Consider the number of times an event occurs over an interval of time and assume
The probability of occurrence is the same for any intervals of equal length
The occurrence in any interval is independent of an occurrence in any non-overlapping interval
If x = the number of occurrences in an interval, then x is a Poisson random variable

7. Poisson Probability Calculations
LO5-4
Poisson Probability Calculations

8. 5.5 The Hypergometric Distribution (Optional)
LO 5-5: Use the hypergeometric distribution to compute probabilities (Optional).
5.5 The Hypergometric Distribution (Optional)
Population consists of N items
r of these are successes
(N-r) are failures
If we randomly select n items without replacement, the probability that x of the n items will be successes is given by the hypergeometric probability formula

9. The Mean and Variance of a Hypergeometric Random Variable
LO5-5
The Mean and Variance of a Hypergeometric Random Variable

10. 5.6 Joint Distributions and the Covariance (Optional)
LO 5-6: Compute and understand the covariance between two random variables (Optional).
5.6 Joint Distributions and the Covariance (Optional)

11. LO5-6
Covariance
To measure the association between x and y, can calculate the covariance between x and y
Calculate (x-µx)(y-µy)=(x-.124)(y-.124) for each combination of values of x and y
Note that .124 is the mean of both distributions
Multiply each (x-µx)(y-µy) value by the probability of p(x,y) and sum results
The result is the covariance
Denoted by σ2xy