Department of Civil Engineering Statistics for Construction Random Variables and Probability distributions Part 3 Prepared by: Dr. Bahram Abedinianagerabi Outline • Triangular Distribution • Poisson Distribution • Normal Distribution Department of Civil Engineering 2 Triangular distribution (Cont’d) • Probability density function (PDF): μ = (a + b + c)/3 Var(x) = (a2+b2+c2 - ab - ac - bc)/18 Department of Civil Engineering 3 Triangular distribution (Cont’d) • Example 6: Duration of a construction project is the followings: Optimistic = 160 days Pessimistic = 185 days Most likely = 170 days Use triangular distribution to answer the following questions: a) The mean and standard deviation: b) What is the probability that the project duration is at least 172 days? c) What is 90 percentile completion time? Solution Department of Civil Engineering 4 Poisson distribution • The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. • The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. • A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, ..., the probability mass function of X. Department of Civil Engineering 5 Poisson distribution (Cont’d) • The Poisson random variable satisfies the following conditions: 1. The number of successes in two disjoint time intervals is independent. 2. The probability of a success during a small time interval is proportional to the entire length of the time interval. Department of Civil Engineering 6 Poisson distribution (Cont’d) • Examples Number of deaths by horse kicking in the Prussian army Birth defects and genetic mutations Rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases Car accidents Traffic flow and ideal gap distance Number of typing errors on a page Hairs found in McDonald's hamburgers Spread of an endangered animal in Africa Failure of a machine in one month Department of Civil Engineering 7 Poisson distribution (Cont’d) e = 2.71828 • The positive real number λ is equal to the expected value of X and also to its variance. Department of Civil Engineering 8 Poisson distribution (Cont’d) • Example 7: In DFW area, snowing occurs once every 500 days on average. a) Calculate the probability of k = 3 snowing days in a 500 day interval, assuming the Poisson model is appropriate. b) Calculate the probability having less than 5 snowing days in a 500 day interval. c) Calculate the probability having equal or more than 3 snowing days in a 500 day interval. Solution Department of Civil Engineering 9 Poisson distribution (Cont’d) • Example 8: Vehicles pass through a junction on a busy road at an average rate of 300 per hour. a) Find the probability that none passes in a given minute. b) What is the expected number passing in two minutes? c) Find the probability that this expected number actually pass through in a given two-minute period. Solution Department of Civil Engineering 10 Normal distribution • Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. • In graph form, normal distribution will appear as a bell curve. Parameters: the mean, , and the standard deviation, . mean = central tendency standard deviation = measure of spread. Department of Civil Engineering 11 Normal distribution • Notation: • Probability density function: Department of Civil Engineering 12 Normal distribution (Cont’d) • Some characteristics of a normal distribution are: Continuous data which has a bell shaped histogram. The MEAN is in the middle. The distribution is symmetrical. A lower mean Total Area = 1 A higher mean x x Department of Civil Engineering x 13 Normal distribution (Cont’d) • Some characteristics of a normal distribution are: 1 Std Dev either side of mean = 68% 2 Std Dev either side of mean = 95% 3 Std Dev either side of mean = 99% Department of Civil Engineering 14 Normal distribution (Cont’d) • Distributions with different spreads have different STANDARD DEVIATIONS A smaller Std Dev. A larger Std Dev. Graph Department of Civil Engineering 15 Normal distribution (Cont’d) • Standard Normal Distribution Properties: Total area under curve = 1 The mean = 0 The standard deviation = 1 The curve is symmetrical P(a < Z < b) = shaded area Department of Civil Engineering 16 Normal distribution (Cont’d) • How to solve problems related to Normal Distributions • The question can be determining the probability of X less or greater than a number Steps: 1) Read the question, Draw a diagram & Shade 2) Convert X to ‘z’ 4) Look up ‘z’ in the tables to get the probability Graphical displays of Normal Distribution Department of Civil Engineering 17 Normal distribution (Cont’d) • Example 9: The heights of the same variety of pine tree are also normally distributed. The mean height is μ=33mm and the standard deviation is 3. Which normal distribution below best summarizes the data? Department of Civil Engineering 18 Normal distribution (Cont’d) • Example 10: Assume that the duration of a construction activity has normal distribution with the mean of 10 days and standard deviation of 1.5 days. a) What is the probability which the activity takes 10 days or less? b) What is the probability which the activity takes 12 days or more? Z table Department of Civil Engineering 19 Normal distribution (Cont’d) • Example 11: The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of 5. a) Find the probability that a randomly selected student scored more than 65 on the exam. b) Find the probability that a randomly selected student scored less than 85. c) Find the 90th percentile (that is, find the score k that has 90% of the scores below k and 10% of the scores above k). d) Find the 70th percentile (that is, find the score k such that 70% of scores are below k and 30% of the scores are above k). Z table Department of Civil Engineering 20 Normal distribution (Cont’d) • Example 12: A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour. a) Find the probability that a household personal computer is used for entertainment between 1.8 and 2.75 hours per day. b) Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment. Z table 2 Department of Civil Engineering 21 Normal distribution (Cont’d) • Example 13: A construction company pays its employees an average wage of $3.25 an hour with a standard deviation of 60 cents. If the wages are approximately normally distributed, determine: a) the proportion of the workers getting wages between $2.75 and $3.69 an hour; b) the minimum wage of the highest 5%. Z table Z table 2 Department of Civil Engineering 22