Download Probability Distributions - Sara McLaughlin Mitchell

Probability Distributions
W&W Chapter 4
Continuous Distributions
Many variables we wish to study in Political
Science are continuous, rather than discrete.
 Military expenditures
 Budget data
We need a continuous probability distribution,
rather than a discrete one (such as the
binomial).
Frequency Bar Graphs
Recall the frequency bar graph for a continuous
measure. We create a certain number of
continuous “cells”, and graph their height at the
cell midpoint.
The height of each bar represents the frequency
(f) of cases in each cell.
Relative Frequency Density
We can also graph relative frequency (f/N) using
a bar graph (Figure 4-3).
It is convenient to change the vertical scale to
relative frequency density, which makes the
total area (the sum of all the areas of the bars)
equal to 1.
Relative Frequency Density
Relative Frequency Density = relative frequency = f/N
cell width
cell width
What happens as N increases?
Because the area remains fixed at 1, the relative
frequency density becomes approximately a curve.
We will call this the density or the probability distribution.
Some Calculus
For continuous random variables, sums are
replaced by integrals, the limiting sum in
calculus.
Pr(a  X  b) =  p(x)dx
Think of the integral taking p(x) times dx, where
dx is a small number so that p(x)dx is the area
of a very thin rectangle.
More Calculus
Similarly, the mean and variance are calculated as
integrals for continuous distributions.
 =  xp(x) dx
2 =  (x-)2p(x) dx
Unlike in the discrete case, a continuous probability
distribution is not the probability of a specific point x.
The probability of a specific point on a continuous
distribution is _____?
More calculus
Rather we are calculating the probability of the
interval between two points (such as a and b).
Important: the area under the entire probability
density function equals one.
A cumulative distribution function (c.d.f) is
defined by Pr(X  x).
The Normal Distribution
For many random variables, the probability
distribution is a bell-shaped curve called the
normal curve or Gaussian curve (in honor of
the German scientist Karl Friedrick Gauss,
1777-1855).
Also we will discover later that most distributions
have approximately normal sampling
distributions as N gets large, making it very
useful in statistics.
The General Normal Distribution
p(x) = [ 1 ] e[-1 (x - )2]
 2 2 2
where
 = mean
2 = variance
e = 2.71828
 = 3.1416
Properties of the Normal
Distribution



As x gets far away from the mean, the negative
exponent decreases p(x) and the probability
approaches zero symmetrically in both tails.
The mean, , is the center point.
The mean, , and the variance, 2 are the
parameters of the normal curve, meaning they
are all you need to know to characterize the
distribution.
The Standard Normal Distribution
p(z) = [ 1 ] e[-1/2(z)2]
 2
The mean and variance of a distribution vary significantly
across different samples, depending on how our
variables are measured. So we need a way to
standardize the distribution so that we can make
comparisons across samples. We do this by
computing a Z-score.
Z-scores
Z = (x - )

Example: IQ's in the US
Suppose the mean IQ is 100 and the standard deviation
is 16.
The z-score for a person who has an IQ of 125 or an IQ
of 85 is:
Z = (125 - 100)/16 = 1.56
Z = (85 - 100)/16 = -.9375
Z-scores continued
Scores above the mean have positive z scores,
while scores below the mean have negative z
scores (the mean itself has a z score of zero).
Each z-value is the number of standard
deviations from the mean because for the
standard normal distribution, =0 and =1.
Z-scores continued
The area under the curve gives us the probability above
or below a certain point (recall that the probability of a
particular value of x is zero!).
What proportion of people have higher IQ's than a person
with an IQ of 125?
We know that Z = 1.56
We go to the table in the back of the book (page 672) and
we can see that the area beyond Z = 1.56 is .059. In
other words, 5.9% of people have higher IQ's.
Z-scores continued
To find a probability less than a particular Z, we use the
symmetrical properties of the curve. The total area
under the curve is 1 and half is above the mean and
half is below.
For an IQ of 85, what % of people have lower IQ's?
Z = -.9375
Since the curve is symmetric, the area below -.9375 must
equal the area above .9375. Area for .94 equals .174,
or 17.4% have lower IQ's.
Interesting property of the Standard
Normal Distribution
68% of all points fall within +/- 1 standard
deviation of the mean, 95% within +/- 2
standard deviations, and 99.7% within +/- 3
standard deviations.
Where do these values come from?
Expected Value
Recall that x
= mean of X
= average of X
= expected value of X, or E(X)
=  xp(x)
We can apply the same formula to a function of a
random variable.
Expected Value continued
Suppose the annual cost of clothing (R) is a
function of the number of girls (X) in the family,
that is:
R = g(X) or more specifically,
R = -100X2 + 300X + 500
Expected Value continued
Number of girls Probability
Clothing Cost
x
p(x)
r = g(x)
0
.14 $500, R=-100(0)2 + 300(0) + 500
1
.39 $700, R=-100(1)2 + 300(1) + 500
2
.36 $700, R=-100(2)2 + 300(2) + 500
3
.11 $500, R=-100(3)2 + 300(3) + 500
______________________________________
Expected Value continued
Assume that the price goes down for the third
child because they can use hand me downs
from their older sisters.
To calculate the mean for R, R, we can use the
following formula:
R = g(x)p(x)
Expected Value continued
We multiply p(x) times g(x) to get:
g(x)p(x)
(.14)($500) = $70
(.39)($700) = $273
(.36)($700) = $252
(.11)($500) = $55
g(x)p(x) = $650
Expected Value continued
Thus the average cost in a family of three will be $650.
See Table 4-5 for the longer calculation.
What we have done with the above formula, is expected
value (also called an expectation).
E[g(X)] = g(x)p(x)
One possible form of the function, g(X) is g(X) = (X - )2.
This becomes:
Expected Value continued
E(X - )2 = (x - )2 p(x)
What is this?
The population variance!
The variance is a kind of expected value, the
expected squared deviation from the mean.
Another Example: betting
Let's suppose that the Boston Red Sox and the
Detroit Tigers are fighting for 1st place in the
American League East. The odds that Boston
will defeat Detroit is 3 to 1. You are offered the
following wager: $120 if Detroit wins and $40 if
Boston wins. What is the expected value of
this wager? Personally how much would you
be willing to pay to play this gamble?
Betting example
Odds of Boston winning, d = 1/3.
p = d/(d+1) or d = p/(1-p)
p = 1/3/(1/3 + 1) = 1/4
x
p(x) v
Boston wins
.25 $40
Boston loses
.75 $120
Betting Example
E(x) = p(x)v = (.25)(40) + (.75)(120)
E(x) = 10 + 90 = $100.
If you were risk neutral, you would be willing to
pay $100, more if you were risk acceptant and
less if you were risk averse.