Econometrics 1: A Review of Statistical Concepts 1.1 - Random Variables, Distribution and Density Functions

Question 1

What type of variables are usually dealt with in Econometrics?

Answer 1

Random continuous variables

Question 2

What are the names of the graphs that depict the probabilities associated with the values taken by random variables

Answer 2

Density & distribution functions

Question 3

What do density & distribution functions do?

Answer 3

They depict the probabilities associated with the values taken by random variables

Question 4

Describe the density function of a random variable

Answer 4

The density function of a random variable 𝑋 is often
denoted 𝑓(𝑥). The x-axis, ‘X’, represents the continuous range of values over which the random variable 𝑋 exists and
the y-axis, ‘f(x)’, is the value of the density function. The probabilities associated with variable 𝑋 are
measured by areas under the density function (under the curve). So, for example, the probability
that the value of 𝑋 will be between the values 𝑎 and 𝑏 is equal to the value of… let’s say area 𝐴, i.e. Area 𝐴
𝑃(𝑎 < 𝑋 < 𝑏) = ∫with upper limit of a and lower limit of b 𝑓(𝑥)𝑑𝑥. Hence calculating probabilities from density functions requires
integration. A feature of random variables and their densities is that the area under the whole density is equal
to unity, i.e. ∫ with upper limit of ∞ and lower limit of -∞ 𝑓(𝑥)𝑑𝑥 = 1. Why is this? Let’s suppose that the variable can only take on values in the range 0 to 10. Then the probability of the variable taking on a value between 0 and 10 has to
be 1 and therefore ∫ with upper limit of 10 and lower limit of 0 𝑓(𝑥)𝑑𝑥 = 1

Question 5

Describe the distribution function of a random variable

Answer 5

Related to the density function is the distribution function, often denoted 𝐹(𝑥), which depicts the
probability that 𝑋 takes on values less than 𝑥, i.e. 𝐹(𝑎) = 𝑃(𝑋 ≤ 𝑎) = ∫*upper bound ‘𝑎’ and lower bound ‘−∞’ *𝑓(𝑥)𝑑𝑥 where 𝑎 is a number. The distribution therefore shows the accumulation of the probabilities associated with values of 𝑋 up to 𝑋 = 𝑎. For the density function drawn above, this will look something like the
following:
Graph with y-axis ‘F(x)’ and x-axis ‘X’ with cubic line. It’s asymptotic to a horizontal dotted line at the top of the graph labelled ‘1’ at y-axis. There’s also a horizontal dotted line lower down in the graph, labelled ‘F(𝑎)’ at y-axis, that meets the line; at that point, there’s a dotted vertical line, labelled ‘𝑎’ at x-axis.
The distribution function point 𝐹(𝑎) represents the area under the density function up to the value
𝑎.