Random Variable (And Discrete Random)

Question 1

What is a random variable?

Answer 1

A random variable is a function that maps or assigns a real number to an outcome of a given sample space.

Question 2

What is a discrete random variable?

Answer 2

It is a random variable whose range is countably finite or countably infinite.

Question 3

What is a continuous random variable?

Answer 3

It is a random variable whose range consists of numbers between a given interval, and this is uncountably infinite.

Question 4

What is the probability mass function of a random variable X?

Answer 4

Let X be a discrete random variable with the following range of finite elements:
R_X = {x₁, x₂, x₃, … , x_n}
Then the probability mass function is given by:
P_X(x_i) = P[X = x_i] , for i = 1, 2, 3, …

Question 5

What is the Cumulative Distribution Function of a Random Variable?

Answer 5

It is a function defined as:
F_X(α) = P(X ≤ α) for all α ∈ R, where R is a set of real numbers.
This means, the CDF of a random variable is a function that describes the probability of the random variable being less than or equal to α.

Question 6

What are the properties of the Cumulative Distribution Function?

Answer 6

For all a ≤ b,
1. P (a < X ≤ b) = F_X(b) - F_X(a)
2. P (a > X) = 1- P (a ≤ X), or 1 - F_X(a)
3. F_X is a non-decreasing function, which means F_X(a) < F_X(b) for a < b