Paul T. von Hippel
Department of Sociology & Initiative in Population Research
Ohio State University
300 Bricker Hall
190 N. Oval Mall
Columbus, OH 43210
von-hippel.1@osu.edu
614 688-3768
379 words
Expected value. An expected value is the long-run average of a RANDOM
VARIABLE X.
More formally, the expected value E(X) is a weighted average of all X’s
possible values. For a DISCRETE VARIABLE, the weights are the PROBABILITIES of the X
values, p(X), and the average is obtained by summation:
E( X ) 

 xp( x)
x  
For a CONTINUOUS VARIABLE, the weights are the PROBABILITY DENSITIES of the X values,
f(X), and the average is obtained by integration:

E( X ) 
 xf ( x)
x  
For certain distributions, the integral or sum fails to converge. In that case the expectation
is undefined. Although undefined expectations are fairly rare, a well-known example
occurs in connection with the Cauchy distribution, which is defined as the ratio of two
standard normal variables.
To illustrate the calculations, consider the flipping of a fair coin. Let X be 1 if the
coin comes up heads, and 0 if it comes up tails. Each value of X has the same probability,
p(X=1)=p(X=0)=.5, so the expected value, using the discrete variable formula, is
E ( X )  1(.5)  0(.5)  .5 .
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Whenever X is a DUMMY VARIABLE, as here, the expected value is the probability that X is
1. When X is an INTERVAL VARIABLE, the expected value is the population mean.
Expectation is a linear operation: the expected value for a weighted sum of two
random variables is just the weighted sum of the expectations,
E (aX  bY )  aE ( X )  bE (Y ) ,
where a, b are the weights and X, Y are the variables. For example, suppose that X and Y
are dummy variables associated with two fair coins, and each variable is 1 if its coin
comes up heads. If both coins are tossed, the expected number of heads is
E(X)+E(Y)=.5+.5=1.
If two variables are independent, then the expectation of their product is the
product of their expectations,
E ( XY )  E ( X ) E (Y ) ,
but this relationship does not hold if the variables X and Y are not independent.
The expectation for a general function g of X is just a weighted average of the
values of g(X). For a discrete variable X, the weights are the probabilities of the X values,
p(X), and the average is obtained by summation:
E ( g ( X )) 

 g ( x) p( x)
x  
For a continuous variable X, the weights are the densities of the X values, f(X), and the
average is obtained by integration:

E ( g ( X )) 
 g ( x) f ( x)
x  
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These formulas extend in a straightforward way to the multivariate case where X
and Y are vectors of random variables, and g is a function, possibly vector-valued, of the
variable X.
PAUL T. VON HIPPEL
References
Rice, John A. (1995). Mathematical statistics and data analysis (2nd ed.).
Belmont, CA: Duxbury Press.
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