This document contains examples and explanations of discrete random variables. It begins with an example of counting the number of people in cars arriving in a town centre, showing the frequencies and relative probabilities. It then discusses two examples involving dice rolls and calculating probabilities of sums. The examples show determining the probability distributions and calculating probabilities of certain outcomes. They illustrate discrete random variables and how to calculate and represent their probabilities.

1. Statistics 1 Discrete Random Variables Section 1
1
DISCRETE RANDOM VARIABLES
Section 1
Choose from the following:
Introduction: Traffic chaos in town centre
Example 4.1: Two tetrahedral dice
Example 4.2: Finding k for a random
variable
End presentation

2. Statistics 1 Discrete Random Variables Section 1
2
No. people per car 1 2 3 4 5 > 5
Frequency 560 240 150 40 10 0
Traffic chaos in town centre
0
100
200
300
400
500
600
0 1 2 3 4 5 6
number of people
frequency

3. Statistics 1 Discrete Random Variables Section 1
3
Traffic chaos in town centre
Number of people /
Outcome r
1 2 3 4 5 > 5
Relative frequency /
Probability P(X = r)
0.56 0.24 0.15 0.04 0.01 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6
r
P(X = r)

4. Statistics 1 Discrete Random Variables Section 1
4
Example 4.1
Two tetrahedral dice, each with faces labelled 1, 2, 3
and 4, are thrown and the random variable X
represents the sum of the numbers on which the dice
fall.
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
(i) Find the probability distribution of X.
(ii) Illustrate the distribution; describe the shape of
the distribution.

5. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
1
16
r 2 3 4 5 6 7 8
P(X = r)

6. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

7. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

8. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

9. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
3
16
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

10. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
2
16
3
16
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

11. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
+ 1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
1
16
2
16
3
16
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

12. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
r
P(X = r )
1
16
2
16
3
16
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)

13. Statistics 1 Discrete Random Variables Section 1
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Example 4.1
1
16
2
16
3
16
4
16
3
16
2
16
1
16
r 2 3 4 5 6 7 8
P(X = r)
(iii) What is the probability that any throw of the dice
results in a value of X which is an odd number?
The probability that X yields an odd number
= + +
2
16
4
16
2
16
= P(X = 3) + P(X = 5) + P(X = 7)
1
2=

14. Statistics 1 Discrete Random Variables Section 1
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Example 4.2
The probability distribution of a random variable X is
given by
P(X = r) = kr for r = 1, 2, 3, 4
P(X = r) = 0 otherwise
(i) Find the value of the constant k.
r 1 2 3 4
P(X = r) k 2k 3k 4k
S P(X = r) = 1  k + 2k + 3k+ 4k = 1
 10k = 1
 k = 0.1

15. Statistics 1 Discrete Random Variables Section 1
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Example 4.2
r 1 2 3 4
P(X = r) 0.1 0.2 0.3 0.4
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
r
P(X = r)

16. Statistics 1 Discrete Random Variables Section 1
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Example 4.2 r 1 2 3 4
P(X = r) 0.1 0.2 0.3 0.4
(a) P(both values of X are the same)
= P (X1 = X2 = 1 or X1 = X2 = 2 or X1 = X2 = 3 or
X1 = X2 = 4)
= P(X1 = X2 = 1) + P(X1 = X2 = 2) + P(X1 = X2 = 3) +
P(X1 = X2 = 4)
= P(X1 = 1) × P(X2 = 1) + P(X1 = 2) × P(X2 = 2)
+ P(X1 = 3) × P(X2 = 3) + P(X1 = 4) × P(X2 = 4)
= (0.1)2 + (0.2)2 + (0.3)2 + (0.4)2
= 0.01 + 0.04 + 0.09 + 0.16
= 0.3
X1 = 1st value
X2 = 2nd value

17. Statistics 1 Discrete Random Variables Section 1
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Example 4.2 r 1 2 3 4
P(X = r) 0.1 0.2 0.3 0.4
(b) P(total of the two values is greater than 6)
= P(X1 + X2 > 6)
= P(X1 + X2 = 7 or 8)
= P(X1 + X2 = 7) + P(X1 + X2 = 8)
= P(X1 = 3) × P(X2 = 4) + P(X1 = 4) × P(X2 = 3)
+ P(X1 = 4) × P(X2 = 4)
= 0.3 × 0.4 + 0.4 × 0.3 + 0.4 × 0.4
= 0.12 + 0.12 + 0.16
= 0.4
X1 = 1st value
X2 = 2nd value