Chapter 6

Discrete Random Variables

Copyright ©2018 McGraw-Hill Education. All rights reserved.

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Chapter Outline

6.1 Two Types of Random Variables

6.2 Discrete Probability Distributions

6.3 The Binomial Distribution

6.4 The Poisson Distribution (Optional)

6.5 The Hypergeometric Distribution (Optional)

6.6 Joint Distributions and the Covariance (Optional)

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6.1 Two Types of Random Variables

Random variable: a variable whose value is a numerical value that is 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 listener rating (on a scale of 1 to 5) in an AccuRating music survey

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

LO6-1: Explain the difference between a discrete random variable and a continuous random variable.

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6.2 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

Called a discrete probability distribution

Notation: Denote the value of the random variable x and the value’s associated probability by p(x)

LO6-2: Find a discrete

probability distribution and compute its mean and standard deviation.

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Discrete Probability Distribution Properties

For any value x of the random variable, p(x) 0

The probabilities of all the events in the sample space must sum to 1, that is…

LO6-2

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Expected Value of a Discrete Random Variable

The mean or expected value of a discrete random variable x is:

m is the value expected to occur in the long run and on average

LO6-2

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Variance

The variance is the average of the squared deviations of the different values of the random variable from the expected value

The variance of a discrete random variable is:

LO6-2

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Standard Deviation

The standard deviation is the square root of the variance

The variance and standard deviation measure the spread of the values of the random variable from their expected value

LO6-2

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6.3 The Binomial Distribution

LO6-3: Use the binomial distribution to compute probabilities.

The binomial experiment characteristics…

Experiment consists of n identical trials

Each trial results in either “success” or “failure”

Probability of success, p, is constant from trial to trial

The probability of failure, q, is 1 – p

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

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Binomial Distribution Continued

For a binomial random variable x, the probability of x successes in n trials is given by the binomial distribution:

n! is read as “n factorial” and n! = n × (n-1) × (n‑2) × … × 1

0! = 1

Not defined for negative numbers or fractions

LO6-3

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LO6-3

Binomial Probability Table

Table 6.4 (a) for n = 4, with x = 2 and p = 0.1

p = 0.1

P(x = 2) = 0.0486

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LO6-3

Several Binomial Distributions

Figure 6.6

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Mean and Variance of a Binomial Random Variable

If x is a binomial random variable with parameters n and p (so q = 1 – p), then

Mean m = n•p

Variance s2x = n•p•q

Standard deviation sx = square root n•p•q

LO6-3

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6.4 The Poisson Distribution (Optional)

LO6-4: Use the Poisson

distribution to compute probabilities (Optional).

Consider the number of times an event occurs over an interval of time or space, and assume that

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 nonoverlapping interval

If x = the number of occurrences in a specified interval, then x is a Poisson random variable

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The Poisson Distribution Continued

Suppose μ is the mean or expected number of occurrences during a specified interval

The probability of x occurrences in the interval when μ are expected is described by the Poisson distribution

where x can take any of the values x = 0,1,2,3, …

and e = 2.71828 (e is the base of the natural logs)

LO6-4

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LO6-4

Poisson Probability Table

Table 6.5

μ = 0.4

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Poisson Probability Calculations

LO6-4

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Mean and Variance of a Poisson Random Variable

If x is a Poisson random variable with parameter m, then

Mean mx = m

Variance s2x = m

Standard deviation sx is square root of variance s2x

LO6-4

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LO6-4

Several Poisson Distributions

Figure 6.9

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6.5 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

LO6-5: Use the hypergeometric distribution to compute probabilities (Optional).

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The Mean and Variance of a Hypergeometric Random Variable

LO6-5

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Hypergeometric Example

Population of six stocks

Four have positive returns

We randomly select three stocks

Find P(x = 2), mean, and variance

LO6-5

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6.6 Joint Distributions and the Covariance (Optional)

LO6-6: Compute and understand the covariance between two random variables (Optional).

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Calculating Covariance

To calculate covariance, calculate:

(x – μx)(y – μy) p(x,y)

for each combination of x and y

Example on prior slide yields –0.0318

A negative covariance says that as x increases, y tends to decrease in a linear fashion

A positive covariance says that as x increases, y tends to increase in a linear fashion

LO6-6

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Four Properties of Expected Values and Variances

If a is a constant and x is a random variable, then μax = aμx

If x1,x2,…,xn are random variables, then μ(x1,x2,…,xn)= μx1 + μx2 + … + μxn

If a is a constant and x is a random variable, then σ2ax = a2σ2x

If x1,x2,…,xn are statistically independent random variables, then the covariance is zero

Also, σ2(x1,x2,…,xn)= σ2×1+ σ2×2+…+ σ2xn

LO6-6

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