Title Edexcel S1 Notes Random Variable Probability Distribution Mean Expected Value Normal Distribution 415.0 KB 52
##### Document Text Contents
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Contents

1  Statistical modelling  6
Statistical modelling ............................................................................................................ 6

Definition ........................................................................................................................................................ 6

2  Representation of sample data  7

Variables ............................................................................................................................. 7
Qualitative variables....................................................................................................................................... 7
Quantitative variables .................................................................................................................................... 7
Continuous variables ...................................................................................................................................... 7
Discrete variables ........................................................................................................................................... 7

Frequency distributions ....................................................................................................... 7
Frequency tables ............................................................................................................................................. 7
Cumulative frequency .................................................................................................................................... 7

Stem and leaf & back-to-back stem and leaf diagrams ....................................................... 8
Comparing two distributions from a back to back stem and leaf diagram. .................................................. 8

Grouped frequency distributions ......................................................................................... 8
Class boundaries and widths .......................................................................................................................... 8

Cumulative frequency curves for grouped data .................................................................. 9

Histograms .......................................................................................................................... 9

3   Mode, mean (and median)  11

Mode ................................................................................................................................. 11

Mean ................................................................................................................................. 11

Coding ............................................................................................................................... 12
Coding and calculating the mean ................................................................................................................. 12

Median .............................................................................................................................. 13

When to use mode, median and mean .............................................................................. 13
Mode ............................................................................................................................................................. 13
Median .......................................................................................................................................................... 13
Mean ............................................................................................................................................................. 13

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Independent events
Definition. A and B are independent ⇔ P(A ∩ B) = P(A) × P(B)
It is also true that P(A | B) = P(A | B′ ) = P(A).
A and B are not linked, they have no effect on each other.

To prove that A and B are independent
first find P(A), P(B) and P(A ∩ B) without assuming that P(A ∩ B) = P(A) × P(B),

second show that P(A ∩ B) = P(A) × P(B).

Note: If A and B are not independent then P(A ∩ B) ≠ P(A) × P(B), and must be found in
another way, usually considering sample spaces and/or Venn diagrams.

Example: A red die and a green die are rolled and the total score recorded.

A is the event ‘total score is 7’, B is the event ‘green score is 6’ and C is the event ‘total
score is 10’.
Show that A and B are independent, but B and C are not independent.

Solution: The events A, B and C are shown on the diagram.

green

6 × × × × × × P(A) = , P(B) =

5 × × × × × × and P(A ∩ B) = from diagram

4 × × × × × × P(A) × P(B) = = P(A ∩ B)

3 × × × × × × ⇒ A and B are independent.

2 × × × × × ×

1 × × × × × × P(B) = , P(C) =

and P(B ∩ C) = from diagram

1 2 3 4 5 6 red P(B) × P(C) = ≠ P(B ∩ C)

⇒ B and C are not independent

Example: A and B are independent events. P(A) = 0⋅5 and P(A ∩ B′) = 0⋅3. Find P(B).

Solution: P(A) = 0⋅5 and P(A ∩ B′) = 0⋅3

⇒ P(A ∩ B) = 0⋅5 – 0⋅3 = 0⋅2

A

C

B

0⋅3

A B

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But P(A ∩ B) = P(A) × P(B)
⇒ 0⋅2 = 0⋅5 × P(B)

⇒ P(B) = = 0⋅4

Exclusive events

Definition. A and B are mutually exclusive

⇔ P(A ∩ B) = 0

i.e. they cannot both occur at the same time

⇒ P(A ∪ B) = P(A) + P(B)

Note: If A and B are not exclusive then P(A ∪ B) ≠ P(A) + P(B), and must be found in
another way, usually considering sample spaces and/or Venn diagrams.

Example: P(A) = 0⋅3, P(B) = 0⋅9 and P(A′∩ B′) = 0⋅1.

Prove that A and B are mutually exclusive.

Solution: A′∩ B′ is shaded in the diagram

⇒ P(A′∩ B′) = 1 – P(A ∪ B)

⇒ P(A ∪ B) = 1 – 0⋅1 = 0⋅9

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

⇒ 0.9 = 0⋅3 + 0⋅6 – P(A ∩ B)

⇒ P(A ∩ B) = 0

⇒ A and B are mutually exclusive.

Number of arrangements

drawn without replacement from the bag. Find the probability that there are 2 Red beads

Solution: These beads can be drawn in any order, RRY, RYR, YRR

⇒ P(RRY or RYR or YRR)

= P(RRY) + P(RYR) + P(YRR)

= 0858.0408
35

16
4

17
5

18
7

16
4

17
7

18
5

16
7

17
4

18
5 ==××+××+×× to 3 S.F.

A B

A B

0⋅1

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Normal Distribution, Z =

The standard normal distribution with mean 0 and
standard deviation 1 has equation

The normal distribution tables allow us to find the
area between Z1 and Z2.

The normal distribution with mean μ and standard deviation σ has equation

Using the substitution z =

dz = dx, Z1 = and Z2 =

= the area under the standard normal curve, which we can find from the tables using

Z1 = and Z2 = .
Thus

x

μ X1 X2

f (x)

x

0 Z1 Z2

φ (z)

z

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Index
accuracy, 41
box plots, 16
class boundaries, 7
correlation, 27

context questions, 44

cumulative frequency, 6
cumulative frequency curves, 8

cumulative probability distribution, 33
discrete uniform distribution, 35

context questions, 46
expected mean, 35
expected variance, 35

exclusive events, 26
expectation. See expected values
expectation algebra, 33
expected values, 33

expected mean, 33
expected variance, 33

frequency distributions, 6
grouped frequency distributions, 7

histograms, 8
context questions, 42

independent events, 25
interquartile range, 13, 19
mean, 10

coding, 11
when to use, 12

median
discrete lists and tables, 13
grouped frequency tables, 14
when to use, 12

mode, 10
when to use, 12

normal distribution, 37
context questions, 46
general normal distribution, 37
standard normal distribution, 37
standardising the variable, proof, 50

outliers, 16
percentiles, 16
probability

diagrams for two dice etc, 23
number of arrangements, 26
rules, 22
Venn diagrams, 22

probability distributions, 32
product moment correlation coefficient, 27

between -1 and +1, 48
coding, 28
interpretation, 28

quartiles
discrete lists and tables, 13
grouped frequency tables, 14

random variables, 32
continuous, 32
discrete, 32

range, 19
regression, 30

context questions, 45
explanatory variable, 30
response variable, 30

regression line, 30
interpretation, 31

regression line and coding
proof, 49

relative frequency, 22
sample spaces, 22
scatter diagrams, 27

line of best fit, 27

skewness, 17
context questions, 43

standard deviation, 19
statistical modelling, 5
statistical models

context questions, 41

stem and leaf diagrams, 7
tree diagrams, 24
variables

continuous variables, 6
discrete variables, 6
qualitative variables, 6
quantitative variables, 6

variance, 19
coding, 20