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© Boardworks Ltd 20061 of 39 © Boardworks Ltd 20061 of 39
A2-Level Maths: Statistics 2for Edexcel
S2.3 Continuous distributions
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Continuous uniform distribution
Continuous uniform distribution
Approximating the binomial using a normal
Approximating the Poisson using a normal
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A random variable X is said to have a continuous uniform distribution (or rectangular distribution) over the interval [a, b] if its probability density function has the form:
( )1
0 otherwise
a x bf x b a
f(x)
The graph of its probability density function is as follows:
x
Continuous uniform distribution
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[ ]E2
a bX
and
Proof of E[X]: The result for E[X] follows immediately from the symmetry of the p.d.f..
[ ] ( )21Var
12X b a
Continuous uniform distribution
Key result: If X has a continuous uniform distribution over the interval [a, b], then
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Continuous uniform distribution
( ) ( )
3 3 31 1
3 3
b
ax b a
b a b a
( ) ( )2 2 2 21 12
3 4b ab a a ab b
[ ] .2 2 21 1b b
a a
E X x dx x dxb a b a
Proof of Var[X]:
( )2 21
3b ab a
So, [ ] ( ) ( )2 2 21 1Var
3 4X b ab a a b
( )2 212
12b ab a
( )( )3 3 2 2 b a b a b ab a as
( )21
12b a
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Example: A random variable Y has a continuous uniform distribution in the interval [2, 8]. Find P(Y < μ + σ).
Continuous uniform distribution
2
a b
Using the formulae for E[X] and Var[X],we get:
( )2 21
12b a
The required probability is P(Y < μ + σ) = P(Y < 5 + √3).This probability is represented by the shaded area.
5 3 2
6
2 85
2
3
( )218 2 3
12
Therefore P(Y < 5 + √3) =3 3
6
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Examination-style question: A random variable X is given by the probability density function f (x), where
( )1
5 15100 otherwise
xf x
Find:
a) E[X] and Var[X]
b) P(7 ≤ X < 10)
Examination-style question
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Solution:X has a uniform distribution over the interval (5, 15).
a)
Examination-style question
[ ]E2
a bX
[ ] ( )21Var
12X b a
b) The p.d.f. for X is shown on the diagram below. The probability we require is shaded.
So, P(7 ≤ X < 10) =3
10
5 1510
2
( )21 115 5 8
12 3
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Note: If X has a uniform distribution over the interval (a, b) then the cumulative distribution function of X is:
( ) ( )
0
P
1
x ab a
x a
F x X x a x b
x b
Continuous uniform distribution
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Approximating the binomial using a normal
Continuous uniform distribution
Approximating the binomial using a normal
Approximating the Poisson using a normal
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Approximating the binomial using a normal
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Calculating probabilities using the binomial distribution can be cumbersome if the number of trials (n) is large.
Consider this example:
Approximating the binomial using a normal
Let the number of left-handed people in the school be X.
Then X ~ B[1200, 0.1].
Introductory example:10% of people in the United Kingdom are left-handed.
A school has 1 200 students. Find the probability that more than 140 of them are left-handed.
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The required probability is
P(X > 140) = P(X = 141) + P(X = 142) + … + P(X = 1200).
As no tables exist for this distribution, calculating this probability by hand would be a mammoth task.
A further problem arises if you attempt to work out one of these probabilities, for example P(X = 141):
( ) . .1200 141 1059141P 141 C 0 1 0 9X
One way forward is to approximate the binomial distribution using a normal distribution.
Approximating the binomial using a normal
Calculators cannot calculatethe value of this coefficient –
it is too large!
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Key result: If X ~ B(n, p) where n is large and p is small, then X can be reasonably approximated using a normal distribution:
X ≈ N[np, npq]
where q = 1 – p.
There is a widely used rule of thumb that can be applied to tell you when the approximation will be reasonable:
Approximating the binomial using a normal
A binomial distribution can be approximated reasonably well by a normal distribution
provided np > 5 and nq > 5.
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Approximating the binomial using a normal
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Continuity correction:
Exact distribution: B(n, p)
P(X ≥ x) P(X ≥ x – 0.5)
P(X ≤ x) P(X ≤ x + 0.5)
Approximating the binomial using a normal
Approximate distribution: N[np, npq]
This 0.5 is called thecontinuity correction
factor.
A continuity correction must be applied when approximating a discrete distribution (such as the binomial) to a continuous distribution (such as the normal distribution).
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Approximating the binomial using a normal
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Introductory example (continued): 10% of people in the United Kingdom are left-handed.
A school has 1 200 students. Find the probability that more than 140 of them are left-handed.
Approximating the binomial using a normal
Solution:
Let the number of left-handed people in the school be X.
Then the exact distribution for X is X ~ B[1200, 0.1].
Since np = 120 > 5 and nq = 1080 > 5 we can approximate this distribution using a normal distribution:
X ≈ N[120, 108].
np npq
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So, P(X > 140) = P(X ≥ 141) → P(X ≥ 140.5)
..
140 5 1201 973
108
Approximating the binomial using a normal
StandardizeN[120, 108]
Using continuity correction
You convert 140.5 to the standard normal distribution
using the formula:
~ [ ].N 0,1X
Z
N[0, 1]
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Approximating the binomial using a normal
Therefore P(X ≥ 140.5) = P(Z ≥ 1.973)
= 1 – Φ(1.973)
= 1 – 0.9758
= 0.0242
So the probability of there being more than 140 left-handed students at the school is 0.0242.
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Example: It has been estimated that 15% of schoolchildren are short-sighted. Find the probability that in a group of 80 schoolchildren there will be
Approximating the binomial using a normal
Solution:
Let the number of short-sighted children in the group be X.
Then the exact distribution for X is X ~ B[80, 0.15].
Since np = 12 > 5 and nq = 68 > 5 we can approximate this distribution using a normal distribution:
X ≈ N[12, 10.2].
a) no more than 15 children that are short-sighted
b) exactly 10 children that are short-sighted.
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a) So P(X ≤ 15) → P(X ≤ 15.5)
..
.
15 5 121 096
10 2
Therefore P(X ≤ 15.5) = P(Z ≤ 1.096)
= Φ(1.096)
= 0.8635
Approximating the binomial using a normal
Using continuity correction
StandardizeN[12, 10.2] N[0, 1]
So the probability that no more than 15 children will be short-sighted is 0.8635.
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b) So P(X = 10) → P(9.5 ≤ X ≤ 10.5)
..
.
9 5 120 783
10 2
Approximating the binomial using a normal
..
.
10 5 120 470
10 2
Using continuity correction
N[12, 10.2]
Standardize
N[0, 1]
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Approximating the binomial using a normal
Therefore P(9.5 ≤ X ≤ 10.5) = P(–0.783 ≤ Z ≤ –0.470)
= P(0.470 ≤ Z ≤ 0.783)
= 0.7832 – 0.6808 = 0.1024
The probability that 10 children will be short-sighted is 0.1024.
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Examination-style question:
A sweet manufacturer makes sweets in 5 colours. 25% of the sweets it produces are red.
The company sells its sweets in tubes and in bags. There are 10 sweets in a tube and 28 sweets in a bag. It can be assumed that the sweets are of random colours.
Examination-style question
a) Find the probability that there are more than 4 red sweets in a tube.
b) Using a suitable approximation, find the probability that a bag of sweets contains between 5 and 12 red sweets (inclusive).
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Solution:
a) Let the number of red sweets in a tube be X.
Then the exact distribution for X is X ~ B[10, 0.25].
This distribution cannot be approximated by a normal but its probabilities are tabulated:
P(X > 4) = 1 – P(X ≤ 4)
= 1 – 0.9219
= 0.0781
So the probability that a tube contains more than 4 red sweets is 0.0781.
Examination-style question
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Solution:
b) Let the number of red sweets in a bag be Y.
Then the exact distribution for Y is Y ~ B[28, 0.25].
This distribution can be approximated by a normal since np = 7 and nq = 21 (both greater than 5):
Y ≈ N[7, 5.25]
P(5 ≤ Y ≤ 12) → P(4.5 ≤ Y ≤ 12.5)
Examination style question
npq
Using continuity correction
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Therefore P(4.5 ≤ Y ≤ 12.5) = P(–1.091 ≤ Z ≤ 2.400)
= P(Z ≤ 2.400) – P(Z ≤ –1.091)
= Φ(2.400) – (1 – Φ(1.091))
= 0.9918 – (1 – 0.8623)
So the probability that a bag will contain between 5 and 12 red sweets is 0.8541.
..
.
12 5 72 400
5 25
..
.
4 5 71 091
5 25
Examination style question
N[7, 5.25] N[0, 1]
= 0.8541
Standardize
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Approximating the Poisson using a normal
Continuous uniform distribution
Approximating the binomial using a normal
Approximating the Poisson using a normal
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Approximating the Poisson using a normal
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Key result: If X ~ Po(λ) and λ is large, then X is approximately normally distributed:
X ≈ N[λ, λ]
There is a widely used rule of thumb that can be applied to tell you when the approximation will be reasonable:
Recall that the mean and variance of a Poisson distribution are equal.
Note: A continuity correction is required because we are approximating a discrete
distribution using a continuous one.
Approximating the Poisson using a normal
A Poisson can be approximated reasonably well by a normal distribution provided λ > 15.
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Example: An animal rescue centre finds a new home for an average of 3.5 dogs each day.
Approximating the Poisson using a normal
a) What assumptions must be made for a Poisson distribution to be an appropriate distribution?
b) Assuming that a Poisson distribution is appropriate:
i. Find the probability that at least one dog is rehoused in a randomly chosen day.
ii. Find the probability that, in a period of 20 days, fewer than 65 dogs are found new homes.
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Solution:a) For a Poisson distribution to be appropriate we would need to
assume the following:
b) i) Let X represent the number of dogs rehoused on a given day. So, X ~ Po(3.5).
P(X ≥ 1) = 1 – P(X = 0)
= 1 – 0.0302 (from tables)
= 0.9698
Approximating the Poisson using a normal
1. The dogs are rehoused independently of one another and at random;
2. The dogs are rehoused one at a time;
3. The dogs are rehoused at a constant rate.
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b) ii) Let Y represent the number of dogs rehoused over a period of 20 days. So, Y ~ Po(3.5 × 20) i.e. Po(70).
As λ is large, we can approximate this Poisson distribution by a normal distribution:
Y ≈ N[70, 70].
P(Y < 65) → P(Y ≤ 64.5)
..
64 5 700 657
70
Approximating the Poisson using a normal
StandardizeN[70, 70] N[0, 1]
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Approximating the Poisson using a normal
P(Y ≤ 64.5) = P(Z ≤ –0.657)
= 1 – Φ(0.657)
= 1 – 0.7445
= 0.2555
So the probability that less than 65 dogs are rehoused is 0.2555.
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Examination-style question: An electrical retailer has estimated that he sells a mean number of 5 digital radios each week.
Examination-style question
a) Assuming that the number of digital radios sold on any week can be modelled by a Poisson distribution, find the probability that the retailer sells fewer than 2 digital radios on a randomly chosen week.
b) Use a suitable approximation to decide how many digital radios he should have in stock in order for him to be at least 90% certain of being able to meet the demand for radios over the next 5 weeks.
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Solution:
a) Let X represent the number of digital radios sold in a week.So X ~ Po(5).
P(X < 2) = P(X ≤ 1)
= 0.0404 (from tables).
Examination-style question
b) Let Y represent the number of digital radios sold in a period of 5 weeks.
So, Y ~ Po(25).
We require y such that P(Y ≤ y) = 0.9.
So the probability that the retailer sells fewer than 2 digital radios in a week is 0.0404.
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Since the parameter of the Poisson distribution is large, we can use a normal approximation:
Y ≈ N[25, 25].
P(Y ≤ y) → P(Y ≤ y + 0.5) (using a continuity correction).
Examination-style question
N[25, 25] N[0, 1]Standardize
The 10% point of a normal is 1.282
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Examination-style question
..
0 5 251 282
5
y So,
.30 91y
So the retailer would need to keep 31 digital radios in stock.
. .5 1 282 24 5y