Discrete Random Variables
•To understand what we mean by a discrete
random variable
•To understand that the total sample space
adds up to 1
•To understand the P(X=x) notation
•To use the P(X=x) notation to solve problems
Discrete Random Variables
Value is from an experiment in the real world
The value is numerical
X is a random variable (e.g. X = Heads on a coin)
x is a particular variable (e.g. 1 head in 2 throws)
P(X=x) would be the probability of throwing 1
head in 2 throws of a coin
Possible outcomes can be shown in a sample
space
Are these Discrete Random
Variables?
The average lifetime of a light bulb
Not discrete, as time is continuous
The number of days in January
No, not variable as there are always 31
The number of moves it takes to win a game of
draughts
Yes, as number of moves are whole numbers
and it varies game by game
Sample Space
3 coins are tossed and the number of heads, X,
are recorded
a) Show the sample space
b) Write down the probability distribution
c) Write down the probability function
Sample space
HHH, THH, HTH, HHT, TTH, THT, HTT, TTT
Probability distribution
3 coins are tossed and the number of heads, X,
are recorded
a) Show the sample space
b) Write down the probability distribution
c) Write down the probability function
Sample space
HHH, THH, HTH, HHT, TTH, THT, HTT, TTT
x
P(X=x)
0
⅛
1
⅜
2
⅜
3
⅛
Note that the
probabilities
add up to 1
Probability distribution
3 coins are tossed and the number of heads, X,
are recorded
a) Show the sample space
b) Write down the probability distribution
c) Write down the probability function
Sample space
HHH, THH, HTH, HHT, TTH, THT, HTT, TTT
x
P(X=x)
0
⅛
1
⅜
P(X=x) = ⅛, for x = 0,3
⅜,
x = 1,2
0,
otherwise
2
⅜
3
⅛
Example
A tetrahedral die is numbered 1,2,3,4. The die
is biased.
P(die landing on any number = k/x where k is a
constant.
a) Find the value of k
b) Write down the probability distribution for X,
the number the die lands on after a single roll
x
1
P(X=x) K/1
K/
2
K/
2
K/ + K/ + K/ = 1
+
1
2
3
4
12k + 6k + 4k + 3k = 1
12
3
K/
3
4
K/
4
25k = 1
12
25k = 12
k = 12/25
Example
A tetrahedral die is numbered 1,2,3,4. The die
is biased.
P(die landing on any number = k/x where k is a
constant.
a) Find the value of k
b) Write down the probability distribution for X,
the number the die lands on after a single roll
x
1
P(X=x)
12/
2
25
12/
3
50
12/
4
75
12/
100