The document explains the construction of a probability mass function (pmf) for the random variable representing the number of tails when tossing a coin twice. It outlines a step-by-step process including creating a sample space, establishing a frequency table, and deriving the probability distribution of outcomes. The final pmf is given in terms of the frequencies corresponding to the number of tails observed.

1. PROBABILITY
MASS
FUNCTION

2. is an expression, rule, or law that
defines a relationship between two
or more variables; denoted as f(x).
WHAT IS A FUNCTION?

3. Construct a function for the
random variable which
𝑋
pertains to the number of
tails in each outcome when
tossing a coin two times.
EXAMPLE:

4. TOSSING A COIN TWO
TIMES
• Step 1: Make a random
experiment by using tree
diagram.

5. TOSSING A COIN TWO
TIMES
• Step 2: Get the Sample Space
S.S. = (HH, HT, TH, TT)
n(S.S.) = 4

6. S.S. X = no. of tails
TT 2
TH 1
HT 1
HH 0
• Step 3: Get the Random Variable (X). X is the number of
tails in each outcome.
RANDOM VARIABLE (X)
x = {0, 1, 2}

7. S.S. X = no. of tails Frequency (f)
TT 2 1
TH, HT 1 2
HH 0 1
∑f= 4
• Step 4: Make a frequency table and find the frequency of
each random variables.
FREQUENCY

8. S.S. X = no. of tails Frequency (f) P(X) = f/ f
∑
TT 2 1 1/4
TH, HT 1 2 2/4 or 1/2
HH 0 1 1/4
∑f= 4 ∑P(X) = 1
• Step 5: Get the Probability Distribution of each outcome in
the Sample Space.
PROBABILITY DISTRIBUTION

9. S.S.
X = no. of
tails
Frequency
(f)
P(X) = f/ f
∑
TT 2 1 1/4
TH, HT 1 2 2/4 or 1/2
HH 0 1 1/4
∑f= 4 ∑P(X) = 1
• Step 6: Construct a function for Probability Distribution of
Random Variable (X)
PROBABILITY MASS FUNCTION (PMF)
𝒇 ( 𝒙 ) =
{
1
4
, 𝑋 = 0 ,
1
2
, 𝑋 = 1