1. •Core Subject Title: Statistics and Probability
•Core Subject Description:
At the end of the course, the students must know how
to find the mean and variance of a random variable, to
apply sampling techniques and distributions, to estimate
population mean and proportion, to perform hypothesis
testing on population mean and proportion, and to perform
correlation and regression analyses on real-life problems.
2. Content:
1. Random Variables and Probability Distributions
2. Normal Distribution
3. Sampling and Sampling Distributions
4. Estimation of Parameters
5. Tests of Hypothesis
6. ENRICHMENT: Correlation and Regression Analyses
7. In how many ways can a coin fall?
In how many ways can two coins
fall simultaneously?
In how many ways can two coins
fall consecutively?
8. Random Variable – It is a numerical quantity that
is assigned to the outcome of an experiment. We
use capital letters to represent a random
variable.
Random Variable
10. Not Like an Algebra Variable
In Algebra a variable, like x, is an unknown value:
Example: x + 2 = 6
In this case we can find that x=4
But a Random Variable is different ...
A Random Variable has a whole set of values ...
... and it could take on any of those values,
randomly.
Example: X = {0, 1, 2, 3}
X could be 0, 1, 2, or 3 randomly.
And they might each have a different probability.
11. Discrete and Continuous Random Variables
A random variable may be classified as discrete
or continuous. A discrete random variable is
one that can assume only a countable number
of values. A continuous random variable can
assume infinite number of values in one or more
intervals.
12. Study the following example:
Discrete Random Variables
Continuous Random
Variables
number of pencils in the box
amount of antibiotics in the
vial
number of soldiers in the
troop
Lifetime of light bulbs in
minute
number of rotten tomatoes
in the basket length of wire ropes
number of defective
flashlights
voltage of radio batteries
13. Example:
Classify the following as discrete or
continuous
1. the number of senators present in the
meeting.
2. the weight of new born babies for the
month of June.
3. the number of ballpens in the box.
4. the capacity of the electrical resistors.
5. the amount of salt needed to bake a loaf
of bread.
15. Example 1:
Supposed two coins are tossed consecutively and we are
interested in the number of heads that will come out. Let us
use H to represent the number of heads that will come out.
Determine the values of the random variable H.
Step 1. List the sample space of the experiment
S= {HH, HT, TH, TT}
16. Step 2. Count the number of heads in each outcome and assign
this number to this outcome.
OUTCOME NUMBER OF HEADS
(Value of H)
HH
HT
TH
TT
The values of the random variable H (number of heads)
in this experiment are 0, 1, and 2
2
1
1
0
17. Example 2
A basket contains 10 ripe and 4 unripe bananas. If three bananas
are taken from the basket one after the other, determine the
possible values of the random variable R representing the
number of ripe bananas.
Step 1. List the sample space of this experiment. Let R
represents the ripe bananas and let U represents the unripe
bananas.
S= {RRR, RRU, RUR, URR, UUR, URU, RUU, UUU}
18. Step 2. Count the number of ripe bananas (R)
OUTCOME NUMBER OF RIPE
BANANAS
(Value of R)
RRR
RRU
RUR
URR
UUR
URU
RUU
UUU
3
2
2
2
1
1
1
0
The Values of the
random variable R
(number of ripe
bananas) in this
experiment are
0, 1, 2, and 3
19. Example 3
A basket contains 10 ripe and 4 unripe bananas. If three bananas
are taken from the basket one after the other, determine the
possible values of the random variable U representing the
number of unripe bananas.
Step 1. List the sample space of this experiment. Let R
represents the ripe bananas and let U represents the unripe
bananas.
S= {RRR, RRU, RUR, URR, UUR, URU, RUU, UUU}
20. Step 2. Count the number of unripe bananas (U)
OUTCOME NUMBER OF
UNRIPE BANANAS
(Value of U)
UUU
UUR
URU
RUU
RRU
RUR
URR
RRR
3
2
2
2
1
1
1
0
The Values of the
random variable U
(number of unripe
bananas) in this
experiment are
0, 1, 2, and 3
22. The probability distribution of a
discrete random variable X is a list of
each possible value of X together with
the probability that X takes that value
in one trial of the experiment.
23. Example 1
A basket contains 10 ripe and 4 unripe bananas. If three bananas
are taken from the basket one after the other, determine the
possible values of the random variable R representing the
number of ripe bananas.
Step 1. List the sample space of this experiment. Let R
represents the ripe bananas and let U represents the unripe
bananas.
S= {RRR, RRU, RUR, URR, UUR, URU, RUU, UUU}
24. Step 2. Count the number of ripe bananas (R)
OUTCOME NUMBER OF RIPE
BANANAS
(Value of R)
RRR
RRU
RUR
URR
UUR
URU
RUU
UUU
3
2
2
2
1
1
1
0
The Values of the
random variable R
(number of ripe
bananas) in this
experiment are
0, 1, 2, and 3
25. Step 3. Construct the frequency distribution of the values of the
random variable R.
Number of Ripe
Bananas
(Values of R)
Number of
Occurrence
(frequency)
3
2
1
0
Total
1
3
3
1
8
26. Step 4. Construct the probability distribution of the random
variable R by getting the probability of occurrence of each
value of the random variable.
Number of Ripe
Bananas
(Values of R)
Number of
Occurrence
(frequency)
Probability
P(R)
3 1
2 3
1 3
0 1
27. Number of Ripe
Bananas
(Values of R)
Probability
P(R)
3
2
1
0
The probability distribution of the random variable R can be written
as follows
29. Example 2
Supposed two coins are tossed consecutively and we are
interested in the number of heads that will come out. Let us
use H to represent the number of heads that will come out.
Determine the values of the random variable H.
Step 1. List the sample space of the experiment
S= {HH, HT, TH, TT}
30. Step 2. Count the number of heads in each outcome and assign
this number to this outcome.
OUTCOME NUMBER OF HEADS
(Value of H)
HH
HT
TH
TT
The values of the random variable H (number of heads)
in this experiment are 0, 1, and 2
2
1
1
0
31. Step 3. Construct the frequency distribution of the values of the
random variable H.
Number of
HEADS
(Values of H)
Number of
Occurrence
(frequency)
2
1
0
Total
1
2
1
4
32. Step 4. Construct the probability distribution of the random variable R
by getting the probability of occurrence of each value of the random
variable.
Number of
HEADS
(Values of H)
Number of
Occurrence
(frequency)
Probability
P(H)
2 1
1 2
0 1
OR
33. Number of
HEADS
(Values of H)
Probability
P(H)
2
1
0
The probability distribution of the random variable H can
be written as follows
OR
35. Properties of a Discrete Probability
Distribution
1. A probability distribution of a discrete random
variable is a correspondence that assigns probabilities to
the values of a random variable. The probability
distribution of a discrete random variable is also called
the probability mass function.
2. Probability distribution is a listing of the possible
values and the corresponding probabilities of a discrete
random variable or a formula for the probabilities.
36. For any discrete random variable X, the
following are true.
0 ≤ P(X) ≤ 1, for each value of X
3. The probability histogram is a bar graph that
displays the possible values of a discrete
random variable
37. Board work:
A pair of fair dice is rolled. Let X denote
the sum of the number of dots on the top
faces.
1.Construct the probability distribution
of X.
2.Find the probability of the sum of values
of tossing two dice.
3.Construct the histogram of the
