Transcript Lecture9

Problems
Problems 3.75, 3.80, 3.87
4. Random Variables
4. Random Variables
Insurance companies have to take risks.
When you buy insurance you are buying it in
case something goes wrong. The insurance
company is securing you and making money
by betting that you are going to live a long
life or that you are not going to crash your
car.
4. Random Variables
Insurance companies have to take risks.
When you buy insurance you are buying it in
case something goes wrong. The insurance
company is securing you and making money
by betting that you are going to live a long
life or that you are not going to crash your
car.
It’s important that the insurance company
offers it’s insurance at a fair price. How do
they calculate it?
4. Random Variables
It’s important that the insurance company
offers it’s insurance at a fair price. How do
they calculate it?
Here is a simple model. An insurance
company offers a death and disability policy
which pay $10,000 when you die or $5000 if
you are disabled. The company charges
$50/year for this benefit. Should the
company expect a profit?
4. Random Variables
Here is a simple model. An insurance
company offers a death and disability policy
which pay $10,000 when you die or $5000 if
you are disabled. The company charges
$50/year for this benefit. Should the
company expect a profit?
We need to define some terms first.
4. Random Variables
A random variable is a way of recording a
numerical result of a random experiment.
Each sample point is given one numerical
value.
4. Random Variables
A random variable is a way of recording a
numerical result of a random experiment.
Each sample point is given one numerical
value.
In this case the random variable is the
payout of the insurance company.
4. Random Variables
A random variable is a way of recording a
numerical result of a random experiment.
Each sample point is given one numerical
value.
In this case the random variable is the
payout of the insurance company.
We usually use capital X to represent our
random variable.
4. Random Variables
In this case the random variable is the
payout of the insurance company.
We usually use capital X to represent our
random variable.
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
4. Random Variables
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
Now we need to know what happens.
4. Random Variables
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
Now we need to know what happens. The
insurance company uses actuaries to
determine the probability of certain events
occurring.
4. Random Variables
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
The probability the insurance company will
have to pay $10,000 is 1 in a thousand is
represented with:
1
P( X  10,000) 
1000
4. Random Variables
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
The probability the insurance company will
have to pay $5,000 is 2 in a thousand is
represented with:
2
P( X  5,000) 
1000
4. Random Variables
In this case the random variable is the
payout of the insurance company.
We usually use capital X to represent our
random variable.
What
Happens:
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
Probability
P(X=x)
4. Random Variables
In this case the random variable is the
payout of the insurance company.
We usually use capital X to represent our
random variable.
What
Happens:
Probability
P(X=x)
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
2/1000
997/1000
1/1000
4. Random Variable
What
Happens:
Probability
P(X=x)
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
2/1000
997/1000
1/1000
This is called a probability distribution table.
4. Random Variable
What
Happens:
Probability
P(X=x)
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
2/1000
997/1000
1/1000
This is called a probability distribution table.
We may draw the distribution (on a
histogram)
Discrete vs Continuous
Random Variables
The random variable in the previous
example is called a discrete random
variable, since X takes an one of a specific
number of values
A continuous random variable is one that
can take on a range of values inside of an
interval. (Example: X represents the height
of a randomly selected individual).
4. Random Variable
What
Happens:
Probability
P(X=x)
Death
X
the company
pays out
$10,000
Disability
Healthy
$5000
$0
2/1000
997/1000
1/1000
Should the company earn be selling these
policies?
Expected value
Mean or expected value of a discrete
random variable is:
µ = E(x) = ∑ x P (x)
Expected value
Mean or expected value of a discrete
random variable is:
µ = E(x) = ∑ x P (x)
The standard deviation of a discrete random
variable is given by    2 where :
   x p( x) 
2
2
2
Example: Concert Planning
• In planning a huge outdoor concert for June 16, the
producer estimates the attendance will depend on the
weather according to the following table. She also finds out
from the local weather office what the weather has been
like, for June days in the past 10 years.
– Weather
Attendance
Relative Frequency
– wet, cold
5,000
.20
– wet, warm
20,000
.20
– dry, cold
30,000
.10
– dry, warm
50,000
.50
– What is the expected (mean) attendance?
– The tickets will sell for $9 each. The costs will be $2 per
person for the cleaning and crowd-control, plus $150,000 for
the band, plus $60,000 for administration (including the
facilities). Would you advise the producer to go ahead with
the concert, or not? Why?
Properties of Probability,
P( X = xi )
(1)
0  P ( X  xi )  1
n
(2)
 P( X  x )  1
i 1
i
Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) .2
.3
.2
.1
.2
Find
P (x≤17)
P (x≥17)
P (x <16 or x >17)
P (x =19)
P(x≤19)
Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) 0.2 0.3
0.2
0.1
0.2
Find
P (X≤17)= .7
P (X≥17) =.3
P (X <16 or X >17)= .5
P (X =19)= .2
P(X≤19)= 1
Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) 0.2 0.3
0.2
0.1
0.2
Find:
The expect value and standard deviation of
this random variable.
Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) 0.2 0.3
0.2
0.1
0.2
Find:
The expect value and standard deviation of
this random variable.
µ = E(X)=16.8
Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) 0.2 0.3
0.2
0.1
0.2
Find:
The expect value and standard deviation of
this random variable.
2
2
µ = E(X)=16.8 and  2 
x p( x)  


Example
The random variable x has the following
discrete probability distribution:
x=
15
16
17
18
19
P(X=x) 0.2 0.3
0.2
0.1
0.2
Find:
The expect value and standard deviation of
this random variable.
  1.96  1.4
µ = E(X)=16.8 and
Empirical Rule and Chebyshev’s
Rule
Chebyshev’s Rule and the Empirical Rule
for Random Variables. That is
1) The number of points that fall within k
standard deviation of the mean is at least:
1-1/k2.
2) If the distribution of the Random Variable
is a normal bell shaped curve, 68% of data
points are in    , 95% are in   2 and
99.7% are in   3 .
Descriptive Phrases
Descriptive Phrases require special care!
–
–
–
–
At most
At least
No more than
No less than
Problems
Problems 4.12, 4.17, 4.36, 4.40, 4.43
Homework
• Review Chapter 3, 4.1-4.3
• Read Chapter 4.4, 5.1-5.3
• Have a great Thanksgiving
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