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QTM1310/ Sharpe
7.1 Expected Value of a Random Variable
The probability model for a particular life insurance policy is
shown. Find the expected annual payout on a policy.
Chapter 7
Random Variables
and
Probability
Models
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7.1 Expected Value of a Random Variable
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Copyright © 2015 Pearson Education. All rights reserved.
7.1 Expected Value of a Random Variable
The probability model for a particular life insurance policy is
shown. Find the expected annual payout on a policy.
A variable whose value is based on the outcome of a
random event is called a random variable.
 1 
 2 
 997 
  $50, 000 
  $0 

 1000 
 1000 
 1000 
  E  X   $100, 000 
If we can list all possible outcomes, the random variable is
called a discrete random variable.
 $200
If a random variable can take on any value between two
values, it is called a continuous random variable.
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We expect that the insurance company will pay out $200 per
policy per year.
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7.2 Standard Deviation of a Random Variable
7.1 Expected Value of a Random Variable
For both discrete and continuous random variables, the set
of all the possible values and their associated probabilities is
called the probability model.
Standard Deviation of a discrete random variable:
 2  Var  X     x     P  x 
2
  SD  X   Var  X 
When the probability model is known, then the expected
value can be calculated:
E  X    x  P  x  (discrete random variable)
E  X  can be written as  but never as x or y
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QTM1310/ Sharpe
7.2 Standard Deviation of a Random Variable
7.3 Properties of Expected Values and Variances
The probability model for a particular life insurance policy is
shown. Find the standard deviation of the annual payout.
Adding a constant c to X:
Multiplying X by a constant a:
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7.2 Standard Deviation of a Random Variable
7.3 Properties of Expected Values and Variances
Example: Book Store Purchases
Suppose the probabilities of a customer purchasing 0, 1, or 2
books at a book store are 0.2, 0.4 and 0.4 respectively.
Addition Rule for Expected Values of Random Variables
What is the expected number of books a customer will
purchase?
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Addition Rule for Variances of (independent) Random
Variables
What is the standard deviation of the book purchases?
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7.2 Standard Deviation of a Random Variable
7.3 Properties of Expected Values and Variances
Example: Book Store Purchases
The expected annual payout per insurance policy is $200 and
the variance is $14,960,000.
  0  0.2   1 0.4   2  0.4 
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If the payout amounts are doubled, what are the new expected
value and variance?
 1.2
 2   x     P  x
2
E  2 X   2 E  X   2  200  $400
= (0 - 1.2) (0.2)  (1 - 1.2) 2 (0.4)  (2 - 1.2) 2 (0.4)
= 0.288 + 0.016 + 0.256 = 0.56
2
Var  2 X   22 Var  X   4 14,960,000  59,840,000
 = 0.56  0.748
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QTM1310/ Sharpe
7.3 Properties of Expected Values and Variances
7.4 Bernoulli Trials
Compare this to the expected value and variance on two
independent policies at the original payout amount.
Bernoulli trials have the following characteristics:
• There are only two outcomes per trial, called success and
failure.
• The probability of success, called p, is the same on every
trial (the probability of failure, 1 – p, is often called q).
E  X  Y   E  X   E Y   2  200  $400
Var  X  Y   Var  X   Var Y   2 14,960,000  29,920,00
• The trials are independent.
Note: The expected values are the same but the variances are different.
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7.3 Properties of Expected Values and Variances
7.4 Bernoulli Trials
The association of two random variables can be measured
using the covariance of X and Y:
Examples of Bernoulli trials include tossing a coin, collecting
yes / no responses from a survey, or shooting free throws in
basketball.
Cov  X , Y   E   X   Y   
When using Bernoulli trials to develop probability models, we
require that trial are independent.
Then, the covariance gives us the extra information needed
to find the variance of the sum or difference of two random
variables when they are not independent:
We can check the 10% Condition: As long as the number of
trials or sample size is less than 10% of the population size,
we can proceed with confidence that the trials are
independent.
Var  X  Y   Var  X   Var Y   2Cov  X , Y 
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7.3 Properties of Expected Values and Variances
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7.5 Discrete Probability Models
The Uniform Model
If X is a random variable with possible outcomes 1, 2, …, n
and P  X  i   1/ n for each i, then we say X has a discrete
Uniform distribution U[1, …, n].
Covariance doesn’t have to be between –1 and +1, which
makes it harder to interpret. To fix this “problem”, we can
divide the covariance by each of the standard deviations to
get the correlation:
Corr  X , Y  
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Cov  X , Y 
When tossing a fair die, each number is equally likely to
occur. So tossing a fair die is described by the Uniform
model
U[1, 2, 3, 4, 5, 6], with P  X  i   1/ 6.
 XY
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7.5 Discrete Probability Models
7.5 Discrete Probability Models
The Geometric Model
Predicting the number of Bernoulli trials required to achieve
the first success.
Example: Closing Sales A salesman normally closes a
sale on 80% of his presentations.
Assuming the presentations are independent:
What model should be used to determine the probability
that he closes his first presentation on the fourth attempt?
What is the probability he closes his first presentation on
the fourth attempt?
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7.5 Discrete Probability Models
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7.5 Discrete Probability Models
The Binomial Model
Predicting the number of successes in a fixed number of
Bernoulli trials.
Example (continued): Closing Sales
What model should be used to determine the probability that
he closes his first presentation on the fourth attempt?
Geometric
What is the probability he closes his first presentation on the
fourth attempt?
P( X  4)  (0.20)3 (0.80)  0.0064
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7.5 Discrete Probability Models
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7.5 Discrete Probability Models
The Poisson Model
Predicting the number of events that occur over a given
interval of time or space. The Poisson is a good model to
consider whenever the data consist of counts of
occurrences.
Example: Professional Tennis
A tennis player makes a successful first serve 67% of the
time. Of the first 6 serves of the next match:
What model should be used to determine the probability that
all 6 first serves will be in bounds?
What is the probability that all 6 first serves will be in bounds?
How many first serves can be expected to be in bounds?
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QTM1310/ Sharpe
7.5 Discrete Probability Models
7.5 Discrete Probability Models
Example (continued): Professional Tennis
Example: Web Visitors
A website manager has noticed that during evening hours,
about 3 people per minute check out from their online
shopping cart and make a purchase. She believes that each
purchase is independent of the others.
What model should be used to determine the probability that
all 6 first serves will be in bounds? Binomial
What is the probability that all 6 first serves will be inbounds?
What probability model would be used to model the number
of purchases per minute?
 6
P( X  6)    (0.67)6 (0.33)0  0.0905
 6
What is the probability that in any one minute at least one
purchase is made?
How many first serves can be expected to be inbounds?
E(X) = np = 6(0.67) = 4.02
What is the probability that no one makes a purchase in the
next 2 minutes?
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7.5 Discrete Probability Models
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7.5 Discrete Probability Models
Example: Satisfaction Survey
A cable provider wants to contact customers to see if they are
satisfied with a new digital TV service. If all customers are in the
452 phone exchange, (so there are 10,000 possible numbers
from 452-0000 to 452-9999):
What probability model would be used to model the selection of a
single number?
What is the probability the number selected will be an even
number?
Example (continued): Web Visitors
What probability model would be used to model the number
of purchases per minute? The Poisson
What is the probability that in any one minute at least one
purchase is made?
P( X  1)  1  P( X  0)
 1
e 3 30
 1  0.0498  0.9502
0!
What is the probability the number selected will end in 000?
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7.5 Discrete Probability Models
7.5 Discrete Probability Models
Example (continued): Satisfaction Survey
Example (continued): Web Visitors
What probability model would be used to model the selection
of a single number? Uniform, all numbers are equally likely.
What is the probability that no one makes a purchase in the
next 2 minutes?
What is the probability the number selected will be
an even number? 0.5
P ( X  0) 
What is the probability the number selected will
end in 000?
0.001
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e 6 30
 0.00248
0!
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QTM1310/ Sharpe
What Have We Learned?
• Probability distributions are still just models.
Foresee the consequences of shifting and scaling
random variables:
• If the model is wrong, so is everything else.
E(X ± c) = E(X) ± c
E(aX) = aE(X)
Var(X ± c) = Var(X)
Var(aX) = a2Var(X)
SD(X ± c) = SD(X)
SD(aX) = |a|SD(X)
• Watch out for variables that aren’t independent.
• Don’t write independent instances of a random variable
with notation that looks like they are the same variables.
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What Have We Learned?
• Don’t forget: Variances of independent random variables
add. Standard deviations don’t.
• Don’t forget: Variances of independent random variables
add, even when you’re looking at the difference between
them.
Understand that when adding or subtracting random
variables the expected values add or subtract well:
E(X ± Y) = E(X) ± E(Y).
However, when adding or subtracting independent random
variables, the variances add:
• Be sure you have Bernoulli trials.
Var(X ±Y) = Var(X) + Var(Y)
Be able to explain the properties and parameters of the
Uniform, the Binomial, the Geometric, and the Poisson
distributions.
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What Have We Learned?
Understand how probability models relate values to
probabilities.
• For discrete random variables, probability models assign a
probability to each possible outcome.
Know how to find the mean, or expected value, of a discrete
probability model and the standard deviation:
E  X    x  P  x
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 =
 x     P x 
2
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