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

3-2 Probability Distributions and Probability Mass Functions Definition

3-3 Cumulative Distribution Functions Definition

Example 3-8

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

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

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.

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.

Example 3-11

3-5 Discrete Uniform Distribution Definition

3-5 Discrete Uniform Distribution Example 3-13

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

3-5 Discrete Uniform Distribution Mean and Variance

3-5 Discrete Uniform Distribution

3-6 Binomial Distribution Random experiments and random variables

3-6 Binomial Distribution Random experiments and random variables

3-6 Binomial Distribution Definition

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

3-6 Binomial Distribution Example 3-18

3-6 Binomial Distribution Example 3-18

3-6 Binomial Distribution Question

3-6 Binomial Distribution Mean and Variance

3-6 Binomial Distribution Example 3-19

3-7 Geometric Distribution Example 3-20

3-7 Geometric Distribution Definition

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

3-7 Geometric Distribution Example 3-21

3-7 Geometric Distribution Definition

3-7 Geometric Distribution Example 3.22

3-9 Poisson Distribution Definition

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

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

3-9 Poisson Distribution Consistent Units

3-9 Poisson Distribution

3-9 Poisson Distribution Example 3-33

3-9 Poisson Distribution Example 3-33

3-9 Poisson Distribution Mean and Variance

End of lecture 2