Activity 2 – Advanced Statistical Concepts and Business Analytics

Chapter 6
Discrete Random Variables

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

1

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)
6-2

2

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.
6-3

3

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.
6-4

4

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
6-5

5

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

6-6

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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
6-7

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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
6-8

8

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
6-9

9

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
6-10

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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

6-11

11

LO6-3
Several Binomial Distributions
Figure 6.6

6-12

12

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

6-13

13

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
6-14

14

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
6-15

15

LO6-4
Poisson Probability Table
Table 6.5

μ = 0.4

6-16

16

Poisson Probability Calculations
LO6-4

6-17

17

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
6-18

18

LO6-4
Several Poisson Distributions
Figure 6.9

6-19

19

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).
6-20

20

The Mean and Variance of a Hypergeometric Random Variable

LO6-5
6-21

21

Hypergeometric Example
Population of six stocks
Four have positive returns
We randomly select three stocks
Find P(x = 2), mean, and variance

LO6-5
6-22

22

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

6-23

23

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
6-24

24

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
6-25

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