1. Cumulative Distribution Function
“Teach A Level Maths” Statistics 1
Cumulative Distribution Function
© Christine Crisp

2. Statistics 1
Edexcel
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3. When we met frequency distributions, we sometimes found the cumulative frequencies.
e.g. 15 12 10 8 5 f 50 35 23 13
Cu. f. 4 3 2 1 x
In a similar way, with a probability distribution, we can find cumulative probabilities.
e.g. 1
0·1
0·2
0·4 5 4 3 2 1 x 1
0·9
0·7
0·3
0·1
We call the distribution of cumulative probabilities the Cumulative Distribution Function.

4. e.g. 1 For a discrete random variable X the p.d.f. is given by
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 4 3 2 1 x
0·25

5. e.g. 1 For a discrete random variable X the p.d.f. is given by
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
0·25 4 3 2 1 x 1
0·75
0·5
0·25

6. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a)

7. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65 1
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -

8. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65 1
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -

9. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -
(b)

10. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65 1
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -
(b)

11. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1 2 3 4 5
F(x)
0·05
0·1
0·35
0·5
0·65 1
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -
(b)

12. e.g. 3. A discrete random variable X has a p.d.f. given by
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.,

13. e.g. 3. A discrete random variable X has a p.d.f. given by
(b) Write out a table showing the probability distribution and the cumulative distribution function.
Solution:
P(X = x) 3 2 1 x k 3k 5k

14. e.g. 3. A discrete random variable X has a p.d.f. given by
(b) Write out a table showing the probability distribution and the cumulative distribution function.
Solution: x 1 2 3
P(X = x)
Now we can add the cumulative probabilities.

15. e.g. 3. A discrete random variable X has a p.d.f. given by
(b) Write out a table showing the probability distribution and the cumulative distribution function.
Solution: x 1 2 3
P(X = x)
F(x)
(c) The cumulative distribution function, F(X) is given by

16. SUMMARY
The Cumulative Distribution Function is given by F(x) where
e.g. If a p.d.f. is defined for x = 1, 2, 3, n
then,

17. Exercise
1. A discrete random variable X has a p.d.f. given by
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.

18. 1. A discrete random variable X has a p.d.f. given by
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,

19. 1. A discrete random variable X has a p.d.f. given by
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) 4k 3
F(X) 5k 3k 2k k
P(X = x) 4 2 1 x

20. 1. A discrete random variable X has a p.d.f. given by
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 1 2 3 4
P(X = x)
F(X)

22. 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.

23. SUMMARY
The Cumulative Distribution Function is given by F(x) where
e.g. If a p.d.f. is defined for x = 1, 2, 3, n
then,

24. e.g. 1 For a discrete random variable X the p.d.f. is given by
Write out a table showing the probability distribution and the cumulative distribution function. Find a formula for F(x), the cumulative distribution function.
Solution: 1
0·75
0·5
0·25 4 3 2 x
The sum of the probabilities is 1, so
where k is a constant.

25. e.g. 2 For a discrete r.v. X the cumulative distribution function F(x) is given in the table.1
0·65
0·5
0·35
0·1
0·05
F(x) 5 4 3 2 x
Find (a) P(X = 4) (b) P(X > 3)
Solution:
(a) -
(b)