STATISTICS and
PROBABILITY
Random Variables and
Probability Distribution
Directions: List down the sample space for the given
experiment
1. Tossing a die
Answer: {1, 2, 3, 4, 5, 6}
2. Tossing 2 coins
Answer: {HH, HT, TT, TH}
3. Tossing a coin and a die
Answer: {1H, 2H, 3H, 4H, 5H, 6H,1T, 2T, 3T, 4T, 5T, 6T}
4. Tossing three coins
Answer:{HHH, HHT, HTH, THH, TTT, TTH,
THT, HTT}
5. Drawing a jack from standard deck of cards
Answer:{J of heart, J of diamond, J of spade, J
of club}
6. Tossing a pair of dice
Answer:{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),…,
(6,6)} – 36 outcomes
7. Drawing a card from standard deck of cards
Answer:{}
8. Spinning a wheel
Answer:{1, 2, 3, 4, 5, 6, 7, 8}
9. Choosing a vowel from the alphabet
Answer:{a, e, i, o, u}
10. Getting a sum of 7 when a pair of dice is
tossed
Answer:{(1,6), (6,1), (2,5), (5,2), (3,4), (4,3)}
Random Variables
Random Variable
Continuous
Discrete
Mean
Variance
Standard
Deviation
A random variable is a variable whose
possible values are determined by chance.
A random variable is typically represented
by an uppercase letter, usually X, while its
corresponding lowercase letter, x, is used to
represent one of its values.
Example: A coin is tossed thrice. Let the
variable X represent the number of heads
that results from this experiment.
Number of Heads (X)
3
2
2
1
2
1
1
0
The value of the variable X can be 0, 1, 2, or 3. Then in this example, X is
a random variable.
A discrete random variable can only take a
finite (countable) number of distinct values.
Distinct values mean values that are exact and
can be represented by nonnegative whole
numbers.
A continuous random variable can assume an
infinite number of values in an interval between
two specific values. This means they can
assume values that can be represented not
only by nonnegative whole numbers but also by
fractions and decimals. These values are often
results of measurement.
Determine if the random variable X or Y is
discrete or continuous:
1. X = number of points scored in the last
season by a randomly selected
Discrete
basketball player in the PBA
2. Y = the heights of a randomly selected
student inside the library in cm Continuous
3. X = number of birds in a nest Discrete
4. Y = the hourly temperature last Continuous
Sunday
Probability Distribution of Discrete Random Variable
A listing of all possible values of a discrete random
variable along with their corresponding probabilities is
called a discrete probability distribution (probability
mass function). The discrete probability distribution can
be presented in tabular, graphical, or formula form.
Properties of Discrete Probability Distribution
Example 1: Determine whether each of the
following is a discrete probability distribution:
a.
x
1
2
3
4
5
P(x)
0.10
0.20
0.25
0.40
0.05
Yes. The probability of each value of a discrete
random variable is between 0 and 1 and the sum
of all probabilities is 1.
b.
x
1
2
3
4
5
P(x)
0.05
0.25
0.33
0.25
0.08
No. Although the probability of each value is
between 0 and 1, the sum of their probabilities is
not equal to 1.
x
1
2
3
4
P(x)
0.21
29c
0.29
0.21
Example 3: Suppose three coins are tossed.
Let X be the number of tails that occur. Find the
probability of each of the values of the random
variable X.
SS = {TTT, TTH, THT, HTT, HHT, HTH, THH,
HHH}
P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8
The probability distribution or the probability
mass function of discrete variable X
x
0
1
2
3
P(x)
1/8
3/8
3/8
1/8
Example 4: The spinner below is divided into
12 sections. Let X be the score where the
arrow will stop (numbered as 1, 2, 3, 4, 5).
a. Find the probability that the arrow will
stop at 1, 2, 3, 4, and 5.
b. Construct the discrete probability
distribution of the random variable X and
its corresponding histogram.
a. P(1) = 1/12
P(2) = 2/12 = 1/6
P(3) = 3/12 = ¼
P(4) = 2/12 = 1/6
P(5) = 4/12 = 1/3
b. The discrete probability distribution is:
x
1
2
3
4
5
P(x)
1/12
1/6
1/4
1/6
1/3
Histogram:
P(x)
5/12
4/12
3/12
2/12
1/12
x
1
2
3
4
5
6