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Variables and Random Variables
A variable is a quantity (such as height, income,
the inflation rate, GDP, etc.) that takes on different
values across individuals, families, nations, months,
quarters, etc. A constant, on the other hand, does
not vary--e.g., the number of heads on a person.
A random variable is a type of variable which
has its value determined at least in part by the
element of chance
Measures of Central Tendency
The mode, median, and mean are measures of the central
tendency of a random variable such as the height of males. If
the statement is made with respect to this variable that “the
mode is 5'10",” it means that most common height (or the
height which occurs with the greatest frequency) among males
is 5'10".
The median is the value of the random variable such that
half the observations are above it and half below it. To say that
“median family income in the U.S. is $38,450" is to say that
half of U.S. households have an income below that figure and
half above it.
 The population mean (symbolized by the Greek letter
µ) is the average value of the variable for the population.
Let m denote the number of observations (corresponding to
the size of the population). Thus, we have:
1 m
   Xi
m i 1
Suppose we want to know the average height of adult males
in the U.S. The practical approach would be to measure a
representative sample (meaning, for example, that basketball
players would not be disproportionately represented in the
sample) of the population rather than the entire population.
That is, we estimate the population mean by calculating a
sample mean ( ). Let n be the number of observations in our
sample. Thus we have:
1 n
X   Xi
n i 1
Measures of Dispersion
Often we are interested in looking at the degree of
dispersion of a random variable about its mean value. That
is, are our observations of adult male height all bunched up
around the mean or do we have wide dispersion about the
mean? The population variance ( 2) is a measure of the
dispersion of a random variable . The variance of random
variable X is defined as:
m
2
1
    Xi   
m i 1
2
If we observe only a representative sample of the
population, then : (1) µ is unknown; and (2) all the Xi ’ s
are not known. Thus, we estimate 2 by substituting 
for µ and summing across our sample observations of X
This is called a sample variance (s2):
2
1
Xi  X 
s 

1  n i 1
Note that we must divide through by n - 1 to obtain an
unbaised estimate of 2 --that is s2 is an unbaised estimator
of 2 if E(s2 ) = 2
n
2
The population standard deviation () is given by the square root of the
population variance ( 2). You can think of as the “average deviation from
the mean.” In the case of male adult height, one would like to see that
measure expressed in inches--hence we take the square root of the variance.
Similarly, the sample standard deviation (s) is given by the square root of
the sample variance (s2).
Probability Distributions
The probability density function of variable X is
constructed such that, for any interval (a, b), the
probability that X takes on a value in that interval is the
total area under the curve between a and b. Expressed in
terms of integral calculus, we have:
b
Pr( a  X  b)   p( X )dX
a
You should be familiar
with this diagram
P(X)
Area under curve
represents probability
a
b
X
The normal distribution is probability density
function which is symmetric about the mean--i.e., the
left-hand side of the distribution is a mirror image of the
right-hand side. The formula for the normal probability
density function is given by:
1
.5[( X   ) /  ]2
p( X ) 
e
 2
The normal distribution
68.27%
95.45%
-2
-


2
A random variable Z is said to be standard
normal if it is normally distributed with mean of zero
or and a variance of 1. If X is normally distributed
with mean µ and variance 2, we abbreviate with the
expression:
X ~ N(, 2)
Thus, the expression used to indicate that the distribution
of Z is standard normal is:
Z ~ N(0, 1)
The standard normal distribution
For example:
If a = 1.93, then Pr(Z  a ) = 0.1093
P(Z)
And Pr(Z  a ) 1 - 0.1093 = 0.8907
0
Pr(Z > a) when Z ~ N(0, 1)
a
Correlation of Random Variables
To say that random variables X and Y are correlated is
to say that changes in X are associated with changes in Y in
the probabilistic or statistical sense. However, this does
not necessarily mean that a change in X was the cause of a
change in Y, or vice-versa. That is, “correlation does not
imply causality.”
Technically speaking, the statement “X and Y are
positively correlated” means that the covariance between
random variables X and Y is positive (or greater than
zero).
1, X > E(X) and Y > E(Y)
2, X < E(X) and Y > E(Y)
3, X < E(X) and Y < E(Y)
4, X > E(X) and Y < E(Y)
Y
2
1
E(Y)
3
0
4
E(X)
X
X and Y are positively correlated random variables
The sample covariance between X and Y
(i.e., our estimate of the covariance when we
do not observe the entire populations of X’s or
Y’s) is given by the following formula (the
“hat” indicates an estimate):

n
1
cov( X , Y ) 
( Xi  X )(Yi  Y )

n  1 i 1
The covariance is positive if above average values of X
tend to be paired with above average values of Y, and vice
versa. The covariance is negative (and hence the variables are
negatively correlated) if below average values of X tend to be
paired with above average values of Y, and vice-versa. The
magnitude of the covariance depends partly on the unit of
measurement. Hence, we cannot depend on the size of the
covariance to give an accurate measure of the strength of the
relationship
The correlation coefficient ( ) is a unit-free
measure of correlation. The sample correlation
coefficient is given by:

cov( X , Y )
ˆ 

sxsy
It will always be the case that:
-1   1.
If  = 1, there is a perfect positive ( linear)
correlation between X and Y. If  = -1, there is a
perfect negative (linear) correlate between X and Y.