1. District Random Variables and Probability Distribution
Islamic University of Gaza
Statistics and Probability for Engineers
(ENGC 6310)
Lecture 2:
District Random Variables and Probability Distribution  
Prof. Dr. Yunes Mogheir
Civil and Environmental Engineering Dept.
First Semester/2019

4. 3-2 Probability Distributions and Probability Mass Functions
Definition

5. 3-3 Cumulative Distribution Functions
Definition

6. Example 3-8

7. Example 3-8
Figure 3-4 Cumulative distribution function for Example 3-8.

8. 3-4 Mean and Variance of a Discrete Random Variable
Definition

9. 3-4 Mean and Variance of a Discrete Random Variable
Figure 3-5 A probability distribution can be viewed as a loading with the mean equal to the balance point. Parts (a) and (b) illustrate equal means, but Part (a) illustrates a larger variance.

10. 3-4 Mean and Variance of a Discrete Random Variable
Figure 3-6 The probability distribution illustrated in Parts (a) and (b) differ even though they have equal means and equal variances.

11. Example 3-11

12. 3-5 Discrete Uniform Distribution
Definition

13. 3-5 Discrete Uniform Distribution
Example 3-13

14. 3-5 Discrete Uniform Distribution
Figure 3-7 Probability mass function for a discrete uniform random variable.

15. 3-5 Discrete Uniform Distribution
Mean and Variance

16. 3-5 Discrete Uniform Distribution

17. 3-6 Binomial Distribution
Random experiments and random variables

18. 3-6 Binomial Distribution
Random experiments and random variables

19. 3-6 Binomial Distribution
Definition

20. 3-6 Binomial Distribution
Figure 3-8 Binomial distributions for selected values of n and p.

23. 3-6 Binomial Distribution
Example 3-18

24. 3-6 Binomial Distribution
Example 3-18

25. 3-6 Binomial Distribution
Question

26. 3-6 Binomial Distribution
Mean and Variance

27. 3-6 Binomial Distribution
Example 3-19

28. 3-7 Geometric Distribution
Example 3-20

29. 3-7 Geometric Distribution
Definition

30. 3-7 Geometric Distribution
Figure 3-9. Geometric distributions for selected values of the parameter p.

32. 3-7 Geometric Distribution
Example 3-21

33. 3-7 Geometric Distribution
Definition

34. 3-7 Geometric Distribution
Example 3.22

35. 3-9 Poisson Distribution
Definition

36. Poisson Distribution
Occurrence of random event in continuous dimension of time and space
Natural disasters
Earthquakes
Request for services (bank, airport, market counters)
Car arrivals at intersection
The probability of x occurrences per unit time
Assumptions
Possible to divide the time interval into very small subintervals so as P( occurrence in each sub-interval is very small )
P(x) in each sub-interval is constant
P( two or more occurrences in sub-interval ) is ignored
Independent

37. 3-9 Poisson Distribution
Applications:
Intervals:
Length
time,
area,
volume
Counts:
particles of contamination in semiconductor
flaws in rolls of textiles,
calls to a telephone exchange,
power outages, and
atomic particles emitted from a specimen

40. 3-9 Poisson Distribution
Consistent Units

41. 3-9 Poisson Distribution

42. 3-9 Poisson Distribution
Example 3-33

43. 3-9 Poisson Distribution
Example 3-33

44. 3-9 Poisson Distribution
Mean and Variance

45. End of lecture 2