The document outlines the objectives and lessons related to random variables and probability distributions, focusing on distinguishing between discrete and continuous random variables. It includes examples, classification exercises, and definitions of key concepts such as sample space, probability distribution, expected value, and variance. Lessons also emphasize practical applications through problems and assignments.

1. 3rd Quarter
Statistics and
Probability

2. Module 1:
Random Variables and
Probability
Distributions

3. OBJECTIVES:
At the end of the lesson, the students are
expected to:
a. illustrate a random variable (discrete and
continuous);
b. distinguish between a discrete and a
continuous random;
c. find the possible values of a random
variable.

5. Let Us Try
Direction:
Classify each random variable
as discrete or continuous.

6. 1. The number of arrivals at an
emergency room between midnight
and 6:00a.m.
2. The weight of a sack of rice
labeled “50 kilograms.”

7. 3. The duration of the next
outgoing telephone call from a
business office.
4. The number of kernels of
popcorn in a five-kilo container.

8. 5. The number of applicants for a
job.

9. • variable is a placeholder for real number values
that can be assigned to it
LESSON 1: RANDOM
VARIABES
• Some examples of variables include X = number of
heads or Y = number of cell phones or Z = running
time to movies.

10. LESSON 1: RANDOM
VARIABES
• Engrossed in some numerals associated with the
outcomes.
• For example, if a coin is tossed twice, the set of all
possible outcomes (S) of the experiment is:
• S = {TT, TH, HT, HH}

11. LESSON 1: RANDOM
VARIABES
• S = {TT, TH, HT, HH}
• Sample space (S) = is a collection or a set of
possible outcomes of a random experiment.
• The subset of possible outcomes of an
experiment is called events.

12. LESSON 1: RANDOM
VARIABES
• A sample space may contain several
outcomes which depends on the experiment.
If it contains a finite number of outcomes,
then it is known as discrete or finite sample
spaces.

13. LESSON 1: RANDOM
VARIABES
• X = {0, 1, 2}.
• Then X is called a random variable.

14. RANDOM
VARIABES
01.

15. RANDOM VARIABES
• A random variable is a variable whose value
is determined by the outcome of a random
experiment.
(upper case) X for the random variable and lower case
x1, x2, x3,... for the values

16. RANDOM VARIABES
• Upper case letter for the random variable
and lower case letter x1, x2, x3,... for the
values

17. Types of Random
Variables

18. Discrete random
variable
said to be a random variable X and it has
a finite number of elements or infinite
but can be represented by whole
numbers.
These values usually arise from
counts

19. Finite sets or countable sets : sets having a
finite/countable number of members.
Examples of finite sets:
• P = {0, 3, 6, 9, …, 99}
• Q = {a: a is an integer, 1 < a < 10}
• A set of all English Alphabets (because it is
countable).

20. Infinite sets or uncountable sets: the number of
elements in that set is not countable and it cannot be
represented in a Roster form.
Examples of infinite sets:
• A set of all whole numbers,
• W= {0, 1, 2, 3, 4,…}
• A set of all points on a line
• The set of all integers

21. Continuous
random variable
said to be a random variable Y and it has
an infinite (unaccountable) number of
elements and cannot be represented by
whole numbers.
These values usually come from
measurements.

22. PRACTICE
A teacher’s record has the following:
(a) scores of students in a 50-item test,
(b) gender,
(c) height of the students.
Classify each whether discrete or continuous variable.

23. ANSWERS
• Scores of students in a 50-item test are a
discrete random variable .
• Gender is also a discrete random variable.
• Height of the students is regarded as a
continuous random variable.

24. PRACTICE MORE
DIRECTION: Give the set of possible values for each random variable.
1. The number of coins that match when three coins are tossed at
once.
2. The number of games in the next World Cup Series (best of four up
to seven games).
3. The amount of liquid in a 12-ounce can of soft drink.
4. The average weight of newborn babies born in the Philippines in a
month.
5. The number of heads in two tosses of coin.

25. ASSIGNMENT
Write the possible values of each random variable. Use a ¼
sheet of paper (YELLOW PAPER).
a. X = number of heads in tossing a coin thrice
b. Y = dropout rate (%) in a certain high school

26. Let Us Practice More
DIRECTION: Match the following with each letter on the
probability line.

27. Let Us Practice More
1. A male student is chosen in a group of 4
where 1 is female.

28. Let Us Practice More
____ 2. If you flip a coin, it will come down
heads.

29. Let Us Practice More
____ 2. If you flip a coin, it will come down
heads.

30. Let Us Practice More
____ 3. It will be daylight in Davao City at
midnight.

31. Let Us Practice More
____ 4. Of the 40 seedlings, only 10 survived.

32. Let Us Practice More
____ 5. The third person to knock on the door
will be a female.

33. • Discrete Probability Distribution is a table
listing all possible values that a discrete
variable can take on, together with the
associated probabilities.
LESSON 2:
PROBABILITYDISTRIBUTIONS

34. • The function f(x) is called a Probability
Density Function for the continuous
random variable X where the total area
under the curve bounded by the x-axis is
equal 1.
LESSON 2: PROBABILITY
DISTRIBUTIONS

35. • Relative frequency of an event is the number of
times the event occurs divided by the total number
of trials. For instance, if you observed 100 passing
cars and found that 23 of them were red, the
relative frequency would be 23/100.
PROBABILITIES AS
RELATIVE FREQUENCY

36. EXAMPLE:
Flip 3 coins at same time.
Let random variable x be the number of
heads showing.
PROBABILITIES AS
RELATIVE FREQUENCY

37. EXAMPLE:
Make a probability distribution for X, sum
of 2 rolled dice.
PROBABILITIES AS
RELATIVE FREQUENCY

38. Example 1
The weight of a jar of coffee selected is a continuous
random variable. The following table gives the weights in
kg of 100 jars of coffee recently filled by the machine. It
lists the observed values of the continuous random
variable and their corresponding frequencies.

39. Find the probabilities for
each weight category?

41. • Let X represent a discrete random variable with the
probability distribution function P(X). Then the
expected value of X denoted by E(X), or μ, is defined
as:
• E(X) = μ = Σ (xi × P(xi))
EXPECTED VALUE OF A RANDOM
VARIABLE

42. • To calculate this, we multiply each possible value of
the variable by its probability, then add the results.
• Σ (xi × P(xi)) = {x1 × P(x1)} + {x2 × P(x2)} + {x3 × P(x3)}
+ ...
• E(X) is also called the mean of the probability
distribution.
EXPECTED VALUE OF A RANDOM
VARIABLE

43. • To calculate this, we multiply each possible value of
the variable by its probability, then add the results.
• Σ (xi × P(xi)) = {x1 × P(x1)} + {x2 × P(x2)} + {x3 × P(x3)}
+ ...
• E(X) is also called the mean of the probability
distribution.
EXPECTED VALUE OF A RANDOM
VARIABLE

44. Complete the table
and find the
expected value of x ?

45. Complete the table
and find the
expected value of x ?

46. • Let X represent a discrete random variable with probability
distribution P(X).
• The variance of X denoted by V(X) or σ^2 is defined as:
• V(X) = Σ[{X − E(X)}^2 × P(X) ]
VARIANCE OF A RANDON
VARIABLE

47. • Let X represent a discrete random variable with probability
distribution P(X).
• The variance of X denoted by V(X) or σ^2 is defined as:
• V(X) = Σ[{X − E(X)}^2 × P(X) ]
VARIANCE OF A RANDON
VARIABLE

48. Complete the table
and find variance?