Transcript 15 Cumulative Distribution Function

“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
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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