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EDEXCEL STATISTICS 1 PROBABILITY
Discrete Uniform Distribution
The Discrete uniform distribution is possibly the easiest of any discrete probability
distribution to deal with. Common examples would be for instance
Let X = the number uppermost on a fair unbiased dice
Let X = the number that a fair unbiased 10 sided spinner lands on x=1,2,…….10
In each case the probability of each outcome is equally likely, with the PDF written
as
P( X  x ) 
1
n
x  1,2,......n
A fairground dartboard is divided into 9 equal sectors, a dart is thrown until it lands
in the board, if this is modelled by a discrete uniform distribution then
Let X= the number of the sector that the dart lands in
x
P(X=x)
1
1
9
2
1
9
3
1
9
4
1
9
5
1
9
6
1
9
7
1
9
8
1
9
9
1
9
(i) Write down E( X ) , Var( X )
(ii) Comment on the suitability of this model
Ok, here’s a tip any question that says “write down” or “state” values should usually
suggest that its either obvious, or there is a shortcut.
In this case because of the symmetry of the distribution it is easily proved- and this is
available in most textbooks eg. ref S1 p163/4 the following general result .
For any discrete uniform distribution where P( X  x ) 
Then E( X ) 
1
n
x  1,2,......n
n 1
(n 1)(n 1)
and Var( X ) 
2
12
(i)
9 1
2
10

2
5
E( X ) 
Titus Salt School - A Teachnet Uk 2008 Project
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EDEXCEL STATISTICS 1 PROBABILITY
Discrete Uniform Distribution
(9 1)(9 1)
12
80

12
20

3
Var ( X ) 
(ii) Whats wrong with this model ? – basically think of common sense reasons as
to why the probability of hitting each number may not be equally likely.
Assumes that the thrower simply aims randomly
A real dart game would depend on the skill of the thrower etc
Furthermore consider this extension to the question;
The fair charges £1 per throw, and pays out (20 x no scored) - 10
(in pence)
Find the average payout per throw
Let
P = payout in pence
Then P=20X – 10
E(P )  E(20 X 10)
 20E( X ) 10
 20  5 10
 90
In other words the fair charges £1 a throw and pays out £ 0.90 on average
This makes use of the following general results for
Expectation and Variance of a linear function of a random variable.
E(aX  b)  aE( X )  b
Var(aX  b)  a2Var( X )
Examples of applying these results can be found in the worked solutions to the
Edexcel exam style questions.
Titus Salt School - A Teachnet Uk 2008 Project
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