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STAT 2000
Sections 6.16.3
P
b bilit
Di t ib ti
Probability Distributions
AgrestiFranklin
From Probability to Probability Distributions
Toss two coins:
S = { HH, HT, TH, TT}
Assign probabilities and count the number of heads
for each outcome.
prob:
¼
¼
¼
¼
S= { HH, HT,
TH, TT}
# heads
2
1
1
0
Another way to display this information is in a table
similar to the frequency tables from Chapter 2.
We
call this a probability distribution.
Probability distribution for # heads in two tosses:
#heads
Probability
x
P(x)
0
¼
1
½
2
¼
This distribution represents all possible outcomes
of two coin tosses, and is considered a population.
We call the number of heads, X, a
random variable
.
A
random variable
is a variable with numerical
values that is associated with outcomes in the
sample space.
In this example, X is a discrete random variable.
(Recall discrete vs. continuous data.)
The probability distribution of a discrete random
variable is a listing of the values of the random
variable, together with the associated probability.
(Recall frequency and relative frequency
tables/distributions.)
Requirements for a Probability Distribution
Similar to the rules for probability, there are two
requirements for a probability distribution.
(1) Every probability must be between 0 and 1.
(2) The sum of all the probabilities must equal 1.
x
P(x)
0
¼
1
½
2
¼
There are many practical applications of probability
distributions, especially in the gambling and
insurance industries.
Suppose you have been given the opportunity to
select one of 500 unmarked envelopes.
One of these envelopes contains $1000.
Ten of these envelopes contain $100.
Twenty of these envelopes contain $10.
The remaining envelopes contain nothing.
What is the probability distribution for X, the amount
you could win?
Winning
Amount:
Probability:
$1000
$100
$10
$0
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Suppose an insurance company sells you a
$50,000 policy.
They may have long term data to
indicate that 70% of the time you will have no
losses, 20% of the time you will have a $25,000
loss, and 10% of the time you will have a $50,000
loss.
(This is a VERY simple example!)
P
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di t ib ti
f
th
t
f l
Probability distribution for the amount of loss:
Amount
of Loss
Probability
$
0
.70
$25,000
.20
$50,000
.10
Insurance companies focus on how much they
will pay out ‘on average’ as opposed to being
interested in an individual payout.
This is called the expected value, or the amount
they ‘expect’ to have to pay.
Just as we found the mean for a frequency
distribution, we can find the mean of a probability
distribution.
It is a weighted average, with the
probabilities serving as the weights.
(Your grade calculation is a weighted average.)
How much does the insurance company expect to
have to pay out on a $50,000 policy?
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 Spring '08
 smith
 Normal Distribution, Probability, Standard Deviation, Probability theory

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