1. Random Variables and Probability Distribution (2)
If we have a distribution table as: X 2 3 7
p(x)
0.5
0.3
0.4
Whose can find P(X=3)

2. Random Variables and Probability Distribution (2)
If we have a distribution table as: X 2 4 6
P(x) -1
1.5
0.3
0.2
Is it a probability distribution?

3. Random Variables and Probability Distribution (2)
We can represent the discrete probability distribution by
1. Tables, from these tables we can calculate mean – variance – and standard deviation.
2. Histograms

4. Random Variables and Probability Distribution (2)
Example 1
Over last 20 days the number of operating rooms used at a hospital are:
On 3 days only 1 room were used, on 5 days 2 rooms were used, on 8 days 3 rooms were used, on 4 days all 4 rooms were used.
Create a probability distribution table and a histogram

5. Random Variables and Probability Distribution (2)
Discrete probability distribution table X
frequency
P(x)
0.00 1 3
0.15 2 5
0.25 8
0.40 4
0.20
 sum 20
Where X is a # of operating rooms used in 1 day

6. Random Variables and Probability Distribution (2)
Histograms

7. Random Variables and Probability Distribution (2)
Mean, variance, standard deviation and expectation of discrete probability distribution
1) Mean
The mean of a random variable with a discrete probability distribution is:

8. Random Variables and Probability Distribution (2)
Variance and standard deviation
Variance of a probability distribution can be calculated by the formula
Standard deviation can be calculated as:

9. Random Variables and Probability Distribution (2)
From the previous table compute mean – variance-standard deviation X
frequency px
X.px
X^2.px 1 3
0.15 2 5
0.25
0.5 8
0.4
1.2
3.6 4
0.2
0.8
3.2
 Sum 20
2.65
7.95

10. Random Variables and Probability Distribution (2)
Mean =
Variance
Standard deviation

11. Random Variables and Probability Distribution (2)
The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable
The formula is

12. Random Variables and Probability Distribution (2)
Example3
A box contain 5 balls, two are numbered 3 , one is numbered 4 and two are numbered 5. The balls are mixed and one is selected at random, after a ball is selected, its number is recorded, and then it is replaced.
1. Construct a probability distribution table.
2. Compute mean, variance, expectation and standard deviation

13. Random Variables and Probability Distribution (2)
A discrete probability distribution table X 3 4 5
p(x)
2/5
1/5

14. Random Variables and Probability Distribution (2)
Expectation =