DISCRETE & RANDOM
VARIABLES
Discrete random variable
A discrete random variable is one which may take on
only a countable number of distinct values such as 0, 1,
2, 3, 4, ...
Discrete random variables are usually (but not
necessarily) counts. If a random variable can take only a
finite number of distinct values, then it must be discrete.
Examples of discrete random variables include the
number of children in a family, the Friday night
attendance at a cinema, the number of patients in a
doctor's surgery, the number of defective light bulbs in a
box of ten.
Relative frequency
Number of flat sides up.
0
1
2
3
4
Score
5
1
2
3
4
The only way to get an estimate of the probability is to
throw the sticks many times and find what we call the:
– relative frequency
–Q
Can be different each time why?
– If the stick is thrown 100 times and the results
are:
– Flat side up: 70 giving an estimate probability of 0.7
– Curved side up: 30 giving an estimate probability of 0.3
Listing possibilities:
have a system
3 sticks
2 sticks
1 stick
f
f
f
f
c
c f
c
c c
f
f
f
f
f
c
f
c f
f
c c
c f
f
c f
c
c c f
c c c
f
f
f
f
f
f
f
c
f
f
c f
f
f
c c
f
c f
f
f
c f
c
f
c c f
f
c c c
c f
f
f
c f
f
c
c f
c f
c f
c c
c c f
f
c c f
c
c c c f
c c c c
Working out probabilities:
Each stick falls independently,
f=0.7
c=0.3
1 x P(f,f,f,f) = 0.7 x 0.7 x 0.7 x 0.7= 0.2401 = 0.74
4 x P(f,f,f,c)= 0.73 x 0.3 x 4=
6 x P(f,f,c,c)=0.72 x 0.32 x 6 =
f
f
f
f
f
f
f
c
f
f
c f
f
f
c c
f
c f
f
f
c f
c
f
c c f
c c c
4 x P(f,c,c,c) = 0.7 x 0.33 x 4 =
3 sticks
f
1 x P(c,c,c,c) = 0.34 =
f
f
f
c f
f
f
f
f
c
c f
f
c
f
c f
c f
c f
f
c c
c f
c c
2 sticks
1 stick
f
f
f
f
c
c f
c
c c
c f
f
c c f
f
c f
c
c c f
c
c c f
c c c f
c c c
c c c c
Relative frequency
Number of flat sides up.
0
1
2
3
4
Score
5
1
2
3
4
The score is an example of discrete random variable.
– Let S stand for score. Capital letters are used for random
variables.
– P(S=3) means ‘the probability that S=3
– P(S=3) = 0.4116
s.
1
2
3
4
5
P(S=s)
Probability
function
0.0756
0.2646
0.4116
0.2401
0.0081
Note s is used for individual values of the random variable S
P(X=x) as a stick/bar graph
Probibality (X=x) Uniforme distribution
0.18
0.16
0.14
P (X=x)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
2
3
4
dice
5
6
TASK
Exercise A – Page 53 & 54
Questions: 1, 2, 3, 5 & 6
Do rest at home.
Mean, variance and standard
deviation
s.
1
P(S=s)
Probability function
0.0756 0.2646
2
3
4
0.4116 0.2401
5
0.0081
If I were to throw 10000 times, I could work out the
mean like the below. Multiply each of my
probabilities by 10000 and then divide by 10000
1 756  2  2646  3  4116  4  2401  5  81
mean 
 2.84
10000
Mean, variance and standard
deviation
s.
1
P(S=s)
Probability function
0.0756 0.2646
2
3
4
0.4116 0.2401
5
0.0081
However, multiplying and dividing by 10000
both top and bottom seems unnecessary
and it is
mean  1 0.0756  2  0.2646  3 0.4116  4  0.2401  5  0.0081  2.84
MEAN of:
Discrete random Variables
Mean S =Σs x P(S=s)
The mean of a random variable is usually
denoted by μ (‘mu’)
Task B1, B2
VARIANCE
x
0
1
2
3
4
P(X=x)
Probability function
0.15
0.25
0.25
0.25
0.1
Mean = 0x0.15 + 1x0.25 + 2x0.25 +
3x0.25 + 4x0.1=1.9
x
0
x-μ
-1.9
(x-μ)2 P(X=x) (x-μ)2x P(X=x)
3.61
0.15
0.5415
1
2
3
4
-0.9
0.1
1.1
2.1
0.81
0.01
1.21
4.41
0.25
0.25
0.25
0.1
0.2025
0.0025
0.3025
0.4410
Variance & Standard deviation
x
0
1
2
3
4
x-μ
-1.9
(x-μ)2 P(X=x) (x-μ)2x P(X=x)
3.61
0.15
0.5415
-0.9
0.81
0.25
0.1
0.01
0.25
1.1
1.21
0.25
2.1
4.41
0.1
Variance = σ2
Standard deviation = σ
0.2025
0.0025
0.3025
0.4410
1.49
1.22
The standard deviation or random variables is normally denoted as σ
TASK
Page 56 question 2
Homework – test yourself