Introduction to Random Variables
Random Variable
A random variable is a mapping
X : Ω→R
that assigns a real number X(ω) to each outcome ω
Getting proper intuitions of X is very important to understand probability. Let us try to understand random variables with an example.
Let the experiment be flipping a coin twice and X be the number of heads. Then
P(X=0) = P({TT}) = 1/4,
P(X=1) = P({HT, TH}) = 1/2 and
P(X=2) = P({HH}) = 1/4.
We summarize the information as follows:
ω (outcome) | P({ω}) | X(ω) |
TT | 1/4 | 0 |
TH | 1/4 | 1 |
HT | 1/4 | 1 |
HH | 1/4 | 2 |
x (number of heads) | P(X = x) |
0 | 1/4 |
1 | 1/2 |
2 | 1/4 |
Note that X denotes a random variable and x denotes a possible value of X.
You can relate a random variable with a feature in a dataset.
The feature in a dataset is a term equivalent to a random variable. However remember that the dataset is not population but the sample of the population. If the dataset has N rows, we can say that we have N realizations of one or more random variable.
Distribution Function and Probability function
Cumulative Distribution Function
The cumulative distribution function (or distribution function), CDF, is the function
FX : R→[0,1] defined by
FX(x) = P(X ≤ x)
The cumulative distribution function of x is the sum of probabilities of all those values of random variables that are less than or equal to x. Every CDF is non-decreasing and right continuous. If X is discrete and is defined for x1, x2, … then CDF of X will be discontinuous at points x1, x2, …
Continuing with the above example, CDF for the experiment of flipping a coin twice.
Could you understand how we derived the above function? Let’s see.
According to the definition of CDF, FX(x) = P(X ≤ x) i.e. CDF of x is the sum of probabilities of all those values of a random variable that are less than or equal to x. Therefore,
FX(0) = P(X=0) = 1/4
FX(1) = P(X=0) + P(X=1) = 1/4+ 1/2 = 3/4
FX(2) = P(X=0) + P(X=1) + P(X=2) = 1
Now note that even if the probabilities are defined at a discrete point, the CDF is defined for all the values. Therefore FX(1.6) = 0.75.
Using this definition we conclude the above function and plot its graph as follows.
A function F mapping real line to [0, 1] is a CDF for some probability measure P if and only if it satisfies the following three conditions:
(i) F is non-decreasing i.e. x1 < x2 implies that F(x1) ≤ F(x2).
(ii) F is normalized: lim x→-∞F(x) = 0 and limx→∞F(x) = 1.
(iii) F is right-continuous, i.e. F(x) = F(x+) for all x, where
F(x+) = limy→x, y>x F(y)
Probability Function or Probability Mass Function (For Discrete R.V.)
For discrete random variables,
We define the probability function or probability mass function for X byfX(x) = P(X = x)
X is discrete if it takes countably many values {x1, x2, …}.
Here, fX(x) ≥ 0 for all x ∈ R and ∑ifX(xi) = 1.
The CDF of X is related to fX by

We will use fX and FX simply as f and F. Try writing and plotting probability function for flipping a coin experiment whose CDF is shown in previous section.
Probability density function (For Continuous R.V.)
A random variable X is continuous if there exists a function fX such that fX(x) ≥ 0 for all x,
and for every a ≤ b,
The function fX is called the probability density function (PDF). We have that
and fX(x) = F’X (x) at all points x at which FX is differentiable.
Suppose that X has a PDF
clearly fX(x) ≥ 0 and ∫fX(x)dx = 1.
A random variable with this density is said to have Uniform (0,1) distribution. The CDF is given by
