Most studies in probability and statistics make a distinction between theoretical objects
and their empirical counterparts. Let’s see how these two relate.
Theoretical
Empirical_
(1)
X – random variable
xi – data or observations on X, i = 1,2,3,...,N
(2)
X ~ f(s) pdf
(3)
μ = E[X] theoretical mean
(4)
σ2 = var(X) theoretical variance
(5)
cov(X, Y) = σX,Y
(6)
corr(X, Y) = ρX,Y
f̂ ( x i ) histogram or frequency function
x
s2 =
x
i
sample mean
N
 (x x)
i
N 1

cov ( X, Y ) 

ˆ X ,Y  corr ( X, Y ) 
2
sample variance
 (x
i
 x )( y i  y)
N 1
 (x  x )( y  y)
 ( x  x )  ( y  y)
i
i
2
i
(7)
Yt = βo + β1Xt + εt
Theoretical regression model
2
i
y t  ˆ o  ˆ 1x t  ˆ t
Estimated regression model
Whenever you read statistical studies, you should be very careful to distinguish between
the theory and the empirical counterparts. There are other objects, in addition to those
shown above.