Random Variable

1.  WHAT IS PROBABILITY?
 Probability is the numerical measure of the likelihood that an event will occur.
OR
 Number of outcomes leading to the event divided by the total number of outcomes
possible.

2.  Probability of any event must be between 0 and 1, inclusively.
 The sum of probabilities of all mutually exclusive and collectively exhaustive event is 1.
𝑷(𝑬) =
𝒏
𝑵
where:
𝑛 = number of outcomes in 𝐸
𝑁 = Total number of outcomes
 Formula:

3.  Terminology
Experiment: A process that produces outcomes.
oIt can have more than one possible outcome
oBut there is be only one outcome per trail
Trail: A repetition of the process.
Elementary Event: Cannot be decomposed or broken down into other events.
Event: It is an outcome of an experiment

4.  Sample Space: The set of all elementary events for an experiment
oCan be described in the following formats
oRoster
oTree diagram
oSet builder notation
oVenn diagram
 Union of Sets: The union of two or more sets contains an instance of each
element of the two sets. UNION OF
SETS

5. Intersection of Sets: The intersection of two sets contains only one instance of those elements which are
common to both the sets.
Mutually Exclusive Events: Events with no common outcomes
Independent Events: Occurrence of one event does not affect the occurrence or non-occurrence of the
other events.
Collectively Exhaustive Events: Contains all elementary events for an experiment.
Complementary Events: All elementary events not in the event ‘A’ are in its complementary event
INTERSECTION
OF SETS
COLLECTIVELY
EXHAUSTIVE
EVENTS
COMPLEMENTARY
EVENTS

6. RANDOM VARIABLE
A variable which contains outcomes of a chance experiment.
OR
A variable that take on different values in the population according to some
random mechanism.
It can be of two types:-
1. Discrete
2. Continuous

7. Probability distribution function
The probability distribution function or probability density function (𝑃𝐷𝐹) of a random variable 𝑋 means
the values taken by that random variable and their associated probabilities.
The probability density function of a discrete random variable is also known as probability mass function.

8. Cumulative distribution function
The cumulative distribution function of a random
variable 𝑋 (deined as 𝐹(𝑋)) is a graph associating all
possible values or the range of all possible values
with 𝑃(𝑋 ≤ 𝑥).

9. Expected value of a random variable
 Let 𝑋 be a discrete random variable with set of possible values 𝐷 and 𝑝𝑚𝑓 𝑝(𝑋).
The expected value or mean value of 𝑋 is denoted by:
𝐸(𝑋) or 𝜇𝑥, is
𝐸(𝑋) = 𝜇𝑥 =
𝑥∈𝐷
𝑥 ∙ 𝑝 𝑥

10. The variance and standard deviation
Let 𝑋 have a 𝑝𝑚𝑓 𝑝(𝑥) and expected value 𝜇. Then the variance of 𝑋 is denoted by:
𝑉(𝑋) (or 𝜎𝑥
2
or 𝜎2
), is
𝑉(𝑋) = = 𝐸[(𝑋 − 𝜇)2]
The standard deviation of 𝑋 is:
𝜎𝑋 = 𝜎𝑋
2
Same means but difference in variability.
𝑥∈𝐷
(𝑥 − 𝜇)2
∙ 𝑝 𝑥

11. SHORTCUT FORMULA FOR
VARIANCE
𝑉 𝑋 = 𝜎2
=
𝑥∈𝐷
𝑥2
∙ 𝑝 𝑥 − 𝜇2
= 𝐸 𝑋2
− 𝐸(𝑋) 2