#Statistics #Random_Variables #Probability #Probability_distribution
#Discrete_random_variable #Probability_mass_function #Variance #Standard_deviation
Random variables are used to assign values to results. Ex: variable X which takes the
value 1 if a coin is heads and 0 if a coin is tails. It's now possible with the results to do
other numerical operations.
Random variables can be:
discrete: distinct / separate value
continuous: any value in interval
Example of a discrete variable of a coin toss
⎧1
X = ⎨
⎩
0
H eads
T ails
Example of a continuous variable, the mass of an animal.
Y  exact mass of random animal selected at the New Orleans Zoo.
Constructing a probability distribution for
random variable
Overview
A probability distribution for a discrete random variable is also called a Probability Mass
Function or PMF.
How to construct one
How to interpret one
Consider the example above. Rolling a 6 side die gives a 1/6 chance of getting one side.
So, the chance to get a value from 1 to 6 is 1/6. The Cumulative distribution function
CDF shows the cumulative probabilities. For ex, if you want to know what's the
probability of getting a 4 or less, you need to sum up all the probabilities from 1 to 4.
Probability models example
Valid discrete probability
distribution model
A valid model has to respect the following conditions
all the probabilities have to add up to 1 or 100%
∑ f (x) = 1
x
no negative values (you cannot have negative probabilities)
f (x) ≥ 0
Example:
Probability with discrete random variable
example
In the following example, what's the percentage that Hugo will reach 4 packs bought in
order to get his card.
Mean (expected value) of a discrete
random variable
The expected value is noted as
E(x)
or as
μx
.
The following example shows how to get the expected value of x given the probability
table below. The 2.1 workouts would mean that in 10 weeks, it's expected to do 21
workouts. In some weeks, it might be 1, in others 4, but the total should come close to 21
in 10 weeks.
The general intuition is the following. If you do X amount of activities with their
probabilities, if it's done a lot of times, the average will fall around a certain value.
So, the formula for multiple events becomes:
E(x) = ∑ X ∗ P (X)
Standard deviation of a discrete random
variable
The variance of a discrete random variable is calculated as:
n
2

s x = ∑ ((X i − μ x )
2
× Px )
i=1
The standard deviation of a discrete random variable is calculated as:

σx
= √s 2
=
x
Sources:
n
∑ ((X i − μ x ) 2 × P x )
⎷
i=1
Statistics and Probability _ Khan Academy
@zedstatistics - YouTube