1. 12.SPECIAL PROBABILITY DISTRIBUTIONS
CONTINUOUS
DISCRETE
1. UNIFORM
1. UNIFORM
CONTINUOUS
2. BINOMIAL
2. NORMAL
3. POISSON

2. 12.0 Special Probability Distribution
Lec f 6
12.0 Special Probability Distribution
12.1 Discrete Probability Distributions
(Uniform Distribution)

3. LEARNING OUTCOMES
At the end of the lesson students are able to
(a) Understand the discrete uniform distribution
(b) Find the mean and variance for discrete uniform distributions.

4. 12.1 Discrete Uniform Distributions.
A discrete random variable X is said to have a uniform distribution , if its probability function has a constant value, p and defined as

5. Example 1
Suppose a fair die is rolled. The number obtained is a random variable X with a uniform distributions as shown in the following probability distribution table.

6. From the table , we find that the probability for all variables are .
Solution x 1 2 3 4 5 6
P(X=x)
From the table , we find that the
probability for all variables are
The uniform distribution can be written as

7. mean variance = E(x2) - [E(x)]2 If X is a discrete random variable
with a uniform distribution , then
mean
variance
= E(x2) - [E(x)]2

8. Example 2
An unbiased spinner , numbered 1,2,3,4 is spun. The random variables X is the number obtained. Draw a probability distribution table and find
the probability distribution function and sketch the graph.
The mean of the probability distribution .
The variance of the probability distribution .

9. probability distribution table is
Solution:
probability distribution table is x 1 2 3 4
P(X=x)

10. probability distribution function
f(x) x

11. c) variance,
= E(x2) - μ2
= (2.5)2
=1.25

12. Example 3
For the following uniform distribution function
Show that n=7.
Calculate the mean and variance
Find P(|x-1|≥1)

13. a) probability distribution table is
Solution:
a) probability distribution table is
P(X=x) 3 2 1 -1 -2 -3 x
n=7

14. = 0
variance,
= E(x2) - μ2
= 4-0
= 4

15. P(|x-1|≥1)
c) P(|x-1| ≥1)
|x-1|≥1
= P(x≤ 0)
+ P(x≥2)
= 1 - P(x=1)
x-1≥1
x-1≤ -1
= 1 -
x≥2
x≤ 0 2 |

16. Example 4
A discrete random variable X has the following probability distribution . x 2 4 6 8 10
P(X=x)
0.2
Find
d) Standard deviation

17. Solution:
= P(X=4)
+ P(X=6)
+ P(X=8)
= 0.2
+ 0.2
+ 0.2
= 0.6

18. = P(X=6)
+ P(X=8)
+ P(X=10)
= 0.2
+ 0.2
+ 0.2
= 0.6

19. = 6
variance,
= E(x2) - μ2
=44 – 36
= 8

20. CONCLUSION
11.1 Discrete Uniform Distributions. x 1 2 3 … n
P(X=x) p