Econometrics 1: A Review of Statistical Concepts 1.2 - 'Moments – Mean and Variance'

Question 1

What do ‘moments’ do in this context?

Answer 1

Describe certain characteristics of a random variable and its density/distribution
function

Question 2

Describe & explain ‘mean’

Answer 2

Mean (or expected value): often denoted as 𝐸(𝑋) or simply 𝜇, this measures the central tendency of
the density function of variable 𝑋. You could think of it as being the point at which the function
is balanced, where there is equal mass on either side of this point. The expected value is given by
𝐸(𝑋) = ∫upper bound ‘∞’ and lower bound ‘-∞’ 𝑥 𝑓(𝑥) 𝑑𝑥 ≡ 𝜇 (1)
For a symmetric distribution this would correspond to the value of 𝑋 for which the density is
symmetric on either side, i.e. the centre point.
*There’s a normal distribution graph showing with a vertical dotted line right in the middle labelled ‘E(X)’ at x-axis.
The function 𝐸(.) is the expectations operator. To calculate an expectation, you take whatever is
inside the bracket and use it to multiply with the density function, then integrate. So, in general
𝐸(𝑔(𝑋)) = ∫upper bound ‘∞’ and lower bound ‘-∞’ *𝑔(𝑥)𝑓(𝑥)𝑑𝑥
and hence why we get (1), when 𝑔(𝑋) = 𝑋.

Question 3

Describe & explain ‘variance’

Include what ‘standard deviation’ is in relation to this

Answer 3

Variance: this is a measure of the dispersion of the density around the mean, i.e. how spread out is
the density function. It is given by
𝑣𝑎𝑟(𝑋) = 𝐸[(𝑋 − 𝜇)^2] = ∫upper bound ‘∞’ and lower bound ‘-∞’ (𝑥 − 𝜇)^2 𝑓(𝑥) 𝑑𝑥 ≡ 𝜎^2 (2)
We can see what the variance implies for a density function:
Normal distribution graph (‘(f(x)’ y-axis and ‘X’ x-axis) with 2 distribution curves; one steeper yet narrower than the other. Narrower one labelled ‘Variance = sigma squared’ and the other ‘Variance = sigma bar squared’. Also, dotted vertical line in middle labelled ‘E(X)’ at x-axis.
Both of these distributions have the same mean, but 𝜎̃^2 > 𝜎2, because the distribution is more
dispersed around the mean.
Another term that we may often refer to is the standard deviation, which
is the square root of the variance, i.e. 𝑠.𝑑. (𝑋) = √𝑣𝑎𝑟(𝑋) = √𝜎^2bottom right𝑋 = 𝜎bottom right𝑋