An Economic/Statistical Application of Integration: Probability Theory Jan 17,01 1
An Economic/Statistical Application of Integration:Probability Theory
What is a random variable?
x is a random variable if it has a known distribution. That is, x is a random variable if œ a and bone can determine the probability that .
Note that x takes specific values (e.g., if x isweight, each of us has a specific weight butweight, in the population, has some distribution.
A well known statistics book, Introduction to the Theory of Statistics (Mood & Graybill) definesa continuous random variable as follows.
The variable X is a one-dimensional, continuous random variable if there exists a function such that in the interval , and the probability that is
.
The function is called a “density function” (or a “probability density function”).Any function, , can serve as a density function as long as
and
.
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Examples of Density Functions
The normal density function
is a well known density function
where F and u are parameters in the density function.
But, be warned that does not have a closed-form solution.
Lets start more simply.
Make up a simple density function.
Note that it is not necessary that .
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Consider the Following Four Density Functions
• Example 1
Assume
and
.
So, is not a density function, but
is.
That is
and
.
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• Example 2
otherwise
Is this a density function?
and
.
So, yes.
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• Example 3
Use Mathematica to graph this function and show how it changes as s and n change.
Why? Recollect that
So
So is a density function.
It is called the Extreme Value Distribution.It is the foundation of logit models of choice.
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• Example 4
Note arctan ( ) = tan-1 ( ).
is called the Cauchy distribution.
If it simplifies to the “Willy/Marshall” distribution
Willy and Marshal weretwo former students.
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Given the density function, , the
.
• For example 1
• For example 2
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• For example 3
but
because
as
therefore
as
so, as
and
.
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Therefore, for the Extreme Value Distribution
.
For example
What is the probability that ?
What is ?It is the median of the EV distribution.
How did I figure out that the median of the EV distribution is ?
Since
at median
by definition of the median.
Simplify by letting
solve for " by taking the ln of each side
but
solve for
.
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• For example 4
In words, zero is the median of the distribution.
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Given the density function , what is the probability that X is lessthan or equal to x, where X is a specific value of x?
Denote this probability
So the probability that is
is called the cumulative density function for x.
We have already calculated the CDF (cumulative density function) for the Extreme ValueDistribution and determined that
What is the CDF for?
• Example 1
otherwise
.
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• Example 2
otherwise
• Example 3
We have already determined that for the Extreme Value Distribution
• Example 4
Trig ;
There is not a closed-form solution for for the normal distribution.That is
does not have a closed-form solution if
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However, given specific values for u and F2, one can numerically solve for any x.
Most statistics books provide tables for x given u = 0 and F2 = 1 (the standard normal).
Note that if
then
because
.
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Now consider measures of central tendency.
Measures of central tendency are ways to describe one aspect of a distribution, f(x).
Three measures of central tendency are:mean (expected value),median, andmode.
The mean (expected value) of any continuous random variable x with distribution f(x) is definedas
.
What does E(x) mean? If one randomly sampled onex, one would not expect it to be E(x). But, if onerandomly sampled N x’s, one would expect theaverage value of the sampled x’s to 6 E(x) as N 6 4.
The median of a continuous random variable x with density function f(x) is defined as u where
.
If a density function has a unique global max, the value of x that max f(x) is called the mode.Loosely speaking, the mode is the most common value for x (remember that since x iscontinuously distributed the probability of observing any specific value of x is zero).
If the distribution is symmetric, mean = median.For some distributions, mean = median = mode: e.g. the normal.
• The mean of (example 1)
is
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• The mean of (example 2)
otherwise
is
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• For the Extreme Value Distribution (example 3)
and
where
.
I know that where ( is the Euler constant (2.577). But I have been unableto derive it analytically. Can you?
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Medians
We have already determined the median for the Extreme Value Distribution (example 3) and theWilly/Marshall distribution (example 4).
If u is the median
For the Extreme Value Distribution
on page 9 we determined that
recollect that
where * is the Euler constant -.577216.
So, mean … median
and .
On page 10 we determined that the median for the Willy/Marshall distribution is 0.
Determine the mode for the first three example distributions?
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One question we often ask about distributions is:How dispersed or spread out are the values of x?
The most common measure of spread is variance.
The Variance is a measure of dispersion around the mean .
Variance
/ expected value of .
Another possible measure of the dispersionaround the mean is .
It can be shown that if is the density function for x and is some function of x
We used a special case of this relationship to get .That is, if
.
We can also use it to get
in this case and
.
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The variance of x if
(example 1)otherwise
is
since (page 15)
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The variance of x if
(example 2)
otherwise
is
since (page 15)
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Can we figure out the variance for the Extreme Value Distribution?
sincewhere * is the Euler constant
.
Luckily for us, someone (?) has already determined that
.
For the normal density function
the variance is .