“Teach A Level Maths”
Statistics 1
Cumulative Distribution
Function
© Christine Crisp
Cumulative Distribution Function
Statistics 1
Edexcel
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Cumulative Distribution Function
When we met frequency distributions, we sometimes
found the cumulative frequencies.
e.g.
x
f
Cu. f.
1
5
5
2
8
13
3
10
23
4
12
35
5
15
50
In a similar way, with a probability distribution, we
can find cumulative probabilities.
e.g. 1
x
1
P ( X  x ) 0·1
P ( X  x ) 0·1
2
0·2
0·3
3
0·4
0·7
4
0·2
0·9
5
0·1
1
We call the distribution of cumulative probabilities
the Cumulative Distribution Function.
Cumulative Distribution Function
e.g. 1 For a discrete random variable X the p.d.f.
is given by
P ( X  x )  k for x  1, 2, 3, 4
where k is a constant.
Write out a table showing the probability distribution
and the cumulative distribution function. Find a
formula for F(x), the cumulative distribution function.
Solution:
The sum of the probabilities is 1, so
k  k  k  k 1
 k  0  25
x
1
P ( X  x ) 0·25
2
0·25
3
0·25
4
0·25
Cumulative Distribution Function
e.g. 1 For a discrete random variable X the p.d.f.
is given by
P ( X  x )  k for x  1, 2, 3, 4
where k is a constant.
Write out a table showing the probability distribution
and the cumulative distribution function. Find a
formula for F(x), the cumulative distribution function.
Solution:
The sum of the probabilities is 1, so
k  k  k  k 1
 k  0  25
x
1
P ( X  x ) 0·25
P ( X  x ) 0·25
2
0·25
0·5
3
0·25
0·75
4
0·25
1
F ( x )  0  25x
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)

Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65 
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65  0  5
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65  0  5
 0  15
(b)
P ( X  3)  1  P ( X  3)
1
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65  0  5
 0  15
(b)
P ( X  3)  1  P ( X  3)
1 05
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65  0  5
 0  15
(b)
P ( X  3)  1  P ( X  3)
1 05
 05
Cumulative Distribution Function
e.g. 3. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k (2 x  1),
x  1, 2, 3
where k is a constant.
(a) Find the value of k.
(b) Write out a table showing the probability
distribution and the cumulative distribution function.
(c) Find a formula for F(x), the cumulative distribution
function.
Solution:
(a) Since X is a r.v.,

 P( X  x)  1
k  3k  5k  1  k 
1
9
Cumulative Distribution Function
e.g. 3. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k (2 x  1),
x  1, 2, 3
(b) Write out a table showing the probability
distribution and the cumulative distribution function.
Solution:
1
k
x
1
2
3
9
P(X = x)
k
3k
5k
Cumulative Distribution Function
e.g. 3. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k (2 x  1),
x  1, 2, 3
(b) Write out a table showing the probability
distribution and the cumulative distribution function.
Solution:
1
k
x
1
2
3
9
P(X = x)
1
9
3
9
5
9
Now we can add the cumulative probabilities.
Cumulative Distribution Function
e.g. 3. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k (2 x  1),
x  1, 2, 3
(b) Write out a table showing the probability
distribution and the cumulative distribution function.
Solution:
1
k
x
1
2
3
9
P(X = x)
F(x)
1
9
1
9
3
9
4
9
5
9
9
9
(c) The cumulative distribution function, F(X) is
given by
x2
F(X ) 
, x  1, 2, 3
9
Cumulative Distribution Function
SUMMARY
The Cumulative Distribution Function is given by
F(x) where
F ( x0 )  P( X  x0 )
e.g. If a p.d.f. is defined for x = 1, 2, 3, . . . n
then,
F (3)  P ( X  3)
 P ( X  1)  P ( X  2)  P ( X  3)
Cumulative Distribution Function
Exercise
1. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k ( x  1),
x  0, 1, 2, 3, 4
where k is a constant.
(a) Find the value of k.
(b) Write out a table showing the probability
distribution and the cumulative distribution
function.
Cumulative Distribution Function
1. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k ( x  1),
x  0, 1, 2, 3, 4
where k is a constant.
(a) Find the value of k.
(b) Write out a table showing the probability
distribution and the cumulative distribution
function.
Solution:
(a) Since X is a random variable,

k  2k  3k  4k  5k  1 
 P( X  x)  1
1
k
15
Cumulative Distribution Function
1. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k ( x  1),
x  0, 1, 2, 3, 4
where k is a constant.
(a) Find the value of k.
(b) Write out a table showing the probability
distribution and the cumulative distribution
function.
Solution:
(b)
x
0
1
2
3
4
P(X = x)
k
2k
3k
4k
5k
F(X)
Cumulative Distribution Function
1. A discrete random variable X has a p.d.f.
given by
P ( X  x )  k ( x  1),
x  0, 1, 2, 3, 4
where k is a constant.
(a) Find the value of k.
(b) Write out a table showing the probability
distribution and the cumulative distribution
function.
Solution:
(b)
x
0
1
2
3
4
P(X = x)
1
15
2
15
F(X)
1
15
3
15
3
15
6
15
4
15
10
15
5
15
15
15
Cumulative Distribution Function
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Cumulative Distribution Function
SUMMARY
The Cumulative Distribution Function is given by
F(x) where
F ( x0 )  P( X  x0 )
e.g. If a p.d.f. is defined for x = 1, 2, 3, . . . n
then,
F (3)  P ( X  3)
 P ( X  1)  P ( X  2)  P ( X  3)
Cumulative Distribution Function
e.g. 1 For a discrete random variable X the p.d.f.
is given by
P ( X  x )  k for x  1, 2, 3, 4
where k is a constant.
Write out a table showing the probability distribution
and the cumulative distribution function. Find a
formula for F(x), the cumulative distribution function.
Solution:
The sum of the probabilities is 1, so
k  k  k  k 1
 k  0  25
x
1
P ( X  x ) 0·25
P ( X  x ) 0·25
2
0·25
0·5
3
0·25
0·75
4
0·25
1
F ( x )  0  25x
Cumulative Distribution Function
e.g. 2 For a discrete r.v. X the cumulative
distribution function F(x) is given in the table.
x
F(x)
0
1
2
3
4
5
0·05
0·1
0·35
0·5
0·65
1
Find (a) P(X = 4)
(b) P(X > 3)
Solution:
(a) P ( X  4)  P ( X  4)  P ( X  3)
 0  65  0  5
 0  15
(b)
P ( X  3)  1  P ( X  3)
1 05
 05