Discrete Random Variables

1. Statistics 1 Discrete Random Variables Section 1
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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
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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
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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
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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