Transcript sbs2e_ppt_ch08

Chapter 8
Random Variables
and
Probability Models
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8.1 Expected Value of a Random Variable
A variable whose value is based on the outcome of a
random event is called a random variable.
If we can list all possible outcomes, the random variable is
called a discrete random variable.
If a random variable can take on any value between two
values, it is called a continuous random variable.
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8.1 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.
When the probability model is known, then the expected
value can be calculated:
E  X    x  P  x
EX 
is sometimes written as
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(discrete random variable)
, but not

x or y
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8.1 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.
We expect that the insurance company will pay out $200 per
policy per year.
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8.2 Standard Deviation of a Random
Variable
Standard Deviation of a discrete random variable:
 2  Var  X     x     P  x 
2
  SD  X   Var  X 
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8.2 Standard Deviation of a Random
Variable
The probability model for a particular life insurance policy is
shown. Find the standard deviation of the annual payout.
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8.2 Standard Deviation of a Random
Variable
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.
What is the expected number of books a customer will
purchase?
What is the standard deviation of the book purchases?
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8.2 Standard Deviation of a Random
Variable
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.
What is the expected number of books a customer will
purchase?
2
2
  x P x

 
= (0 - 1.2)2 (0.2)  (1 - 1.2)2 (0.4)  (2 - 1.2)2 (0.4)
= 0.288 + 0.016 + 0.256 = 0.56
What is the standard deviation of the book purchases?
 = 0.56  0.748
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8.3 Properties of Expected Values and
Variances
Adding a constant c to X:
Multiplying X by a constant a:
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8.3 Properties of Expected Values and
Variances
Addition Rule for Expected Values of Random Variables
Addition Rule for Variances of (independent) Random
Variables
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8.3 Properties of Expected Values and
Variances
The expected annual payout per insurance policy is $200 and
the variance is $14,960,000. If the payout amounts are doubled,
what are the new expected value and variance?
E  2 X   2 E  X   2  200  $400
Var  2 X   22 Var  X   4 14,960,000  59,840,000
Compare this to the expected value and variance on two
independent policies at the original payout amount.
E  X  Y   E  X   E Y   2  200  $400
Var  X  Y   Var  X   Var Y   2 14,960,000  29,920,00
Note: The expected values are the same but the
variances are different.
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8.4 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].
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.
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8.4 Discrete Probability Models
Bernoulli Trials
Definition: A Bernoulli Trial is a trial with the following
characteristics:
1) There are only two possible outcomes (success and
failure) for each trial.
2) The probability of success, denoted p, is the same for
each trial. The probability of failure is q = 1 – p.
3) The trials are independent.
The next two probability models apply to experiments with
Bernoulli trials.
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8.4 Discrete Probability Models
The Geometric Model
Predicting the number of Bernoulli trials required to achieve
the first success.
The 10% Condition: For finite samples, the sample should be no more than 10% of the population.
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8.4 Discrete Probability Models
The Geometric Model
Find the mean (expected value) of a random variable X, using a
geometric distribution with probability of success, p.
X
0
1
P(X)
q
p
E(X) = 0q + 1p = p
Var(X) = (0 – p)2q + (1 – p)2p
= p2q + q2p
= pq(p + q) = pq(1)
= pq
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8.4 Discrete Probability Models
Independence
Bernoulli Trials must be independent.
The 10% Condition: From finite populations, the sample should
be no more than 10% of the population.
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8.4 Discrete Probability Models
The Binomial Model
Predicting the number of successes in a fixed number of
Bernoulli trials.
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8.4 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.
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8.4 Discrete Probability Models
For example, a website averages 4 hits per minute. Find the
probability that there will be no hits in the next minute.
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8.4 Discrete Probability Models
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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8.4 Discrete Probability Models
Example (continued): 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?
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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8.4 Discrete Probability Models
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 inbounds?
How many first serves can be expected to be inbounds?
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8.4 Discrete Probability Models
Example (continued): 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? Binomial
What is the probability that all 6 first serves will be inbounds?
 6
P( X  6)    (0.67)6 (0.33)0  0.0905
 6
How many first serves can be expected to be inbounds?
E(X) = np = 6(0.67) = 4.02
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8.4 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?
What is the probability the number selected will
end in 000?
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8.4 Discrete Probability Models
Example (continued): 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? Uniform, all numbers are equally likely.
What is the probability the number selected will be
an even number?
0.5
What is the probability the number selected will
end in 000?
0.001
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8.4 Discrete Probability Models
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 probability model would be used to model the number
of purchases per minute?
What is the probability that in any one minute at least one
purchase is made?
What is the probability that no one makes a purchase in the
next 2 minutes?
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8.4 Discrete Probability Models
Example (continued): 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 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? 0.9502
P( X  1)  1  P( X  0)
e3 30
 1
 1  0.0498  0.9502
0!
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8.4 Discrete Probability Models
Example (continued): 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 is the probability that no one makes a purchase in the
next 2 minutes?
e6 30
P( X  0) 
 0.00248
0!
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• Probability distributions are still just models.
• If the model is wrong, so is everything else.
• 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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• 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.
• Be sure you have Bernoulli trials.
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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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What Have We Learned?
Foresee the consequences of shifting and scaling random
variables, specifically understand the Law of Large Numbers
and that the common understanding of the “Law of
Averages” is false.
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)
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What Have We Learned?
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:
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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