This document provides an introduction to random variables and probability distributions. It defines key terms like random variables, sample space, discrete and continuous random variables, and probability mass functions. For a discrete random variable, the probability distribution takes the form of a probability mass function (PMF) that assigns a probability to each possible value. The PMF is like a theoretical relative frequency distribution. Examples are provided to illustrate calculating probabilities of events for experiments like coin tosses and dice rolls.

1. Module 1 - Lesson 1
Random variables and Probabilty
distribution
Statistics and Probability
Jose Karlo Trocio

2. Guide Questions
● What is random variables?
● What is difference between discrete and
continuous random variable?
● How do we construct probabilty mass function
of a discrete probabilty distribution?

3. Probability is a measure of chance declared through
reasonable opinion.
Given a random experiment, the set of all possible outcomes is called its sample space. Each
outcome may be referred to as a sample point.
Example: In the experiment of tossing a coin, the sample space is the set
S = {HEADS,TAILS}
and HEADS and TAILS are its two sample points.
In rolling a die, the sample space is
S ={1,2,3,4,5,6}
The event E “odd outcomes” is given by
E ={1,3,5}.

4. Let the sample size of an experiment be |S| and the event size be |E|.
Then,
P(E)=
|E|
|S|
In rolling a die, what is
the probability that the
outcome is odd?

5. Let the sample size of an experiment be |S| and the event size be |E|.
Then,
P(E)=
|E|
|S|

6. Let the sample size of an experiment be |S| and the event size be |E|.
Then,
P(E)=
|E|
|S|

7. Let the sample size of an experiment be |S| and the event size be |E|.
Then,
P(E)=
|E|
|S|

8. Let the sample size of an experiment be |S| and the event size be |E|.
Then,
P(E)=
|E|
|S|

10. Welcome to our Math Mystery Escape Room!!
Here, you will find that Venus has a beautiful
name and is the second planet from the Sun. It’s
terribly hot, even hotter than Mercury, and its
atmosphere is extremely poisonous
About the introduction
Progress:
1
3
2
2
2
1
1
1
0
1
8
1
8
3
8
3
8
Thus, the values of random
variable X are {0,1, 2, & 3}

11. PROBABILITY DISTRIBUTION
The set of all possible values of the random
variable X, together with their corresponding
associated probabilities , form the probability
distribution of X, If X is a discrete random variable,
the probability distribution is called a probability
mass function or pmf . The pmf is the theoretical
counterpart of a relative frequency distribution for a
sample data set. The pmf of X is an actual function
P (X=x) . It may be expressed in tabular, graphical,
or formula form.

12. PROBABILITY DISTRIBUTION
Probability Histogram.
Probability Histogram of the Number of Heads
in Tossing Three Coins
0 1 2 3
3
8
2
8
1
8

13. PROBABILITY DISTRIBUTION
We may also define the
above probability mass
function in formula form:
𝑃𝑥 𝑥 = 3
8
,𝑖𝑓 𝑥=1,2
1
8
, 𝑖𝑓 𝑥=0,3

14. NOT A DISCRETE RANDOM VARIABLE

15. DISCRETE RANDOM VARIABLE

16. She is a very funny Turkish girl. She enjoys reading, playing
with her dogs and visiting her partner every weekend. She
is specially good in psychology and mathematics as she
has a very analytical mind. Her friends describe her as loyal
and loving
This is ana
Progress:

17. She is a very funny Turkish girl. She enjoys reading, playing
with her dogs and visiting her partner every weekend. She
is specially good in psychology and mathematics as she
has a very analytical mind. Her friends describe her as loyal
and loving
This is ana
Thus, the values of random variable Y are {0,1, 2, 3, 4, & 5}

18. 1
1.2
0
1
2
3
4
5
6
0 1 2 3 4 5
The probability histogram for this expirement is :

19. Let US SUM IT UP
• A random variable is a discrete random variable if its set of possible
outcomes is countable , such as number of defective chairs produced
in a factory
• A random variable is continuous random variable if it takes a
continuous scale such as heights, weights, and temperature.
Properties of a Probability Distribution
1. The probability of each value of x is a value between 0 and 1.
2. The sum of the probabilities of a value is equal to 1.In symbol,
ΣP(X) = 1.