STATISTICS AND PROBABILITY IN CIVIL ENGINEERING TS4512 Doddy Prayogo, Ph.D.

Discrete Probability Distributions The Most Common Discrete Probability Distributions 1) --- Bernoulli distribution 2) --- Binomial 3) --- Geometric 4) --- Poisson

Bernoulli distribution The Bernoulli distribution is the “coin flip” distribution. X is Bernoulli if its probability function is: X=1 is usually interpreted as a “success.” E.g.: X=1 for heads in coin toss X=1 for male in survey X=1 for defective in a test of product X=1 for “made the sale” tracking performance

Bernoulli distribution Expectation: Variance:

Binomial distribution The binomial distribution is just n independent Bernoullis added up. Testing for defects “with replacement.” Have many light bulbs Pick one at random, test for defect, put it back If there are many light bulbs, do not have to replace

Binomial distribution Let’s figure out a binomial r.v.’s probability function. Suppose we are looking at a binomial with n=3. We want P(X=0): Can happen one way: 000 (1-p)(1-p)(1-p) = (1-p)3 We want P(X=1): Can happen three ways: 100, 010, 001 p(1-p)(1-p)+(1-p)p(1-p)+(1-p)(1-p)p = 3p(1-p)2 We want P(X=2): Can happen three ways: 110, 011, 101 pp(1-p)+(1-p)pp+p(1-p)p = 3p2(1-p) We want P(X=3): Can happen one way: 111 ppp = p3

Binomial distribution So, binomial r.v.’s probability function

Binomial distribution Typical shape of binomial: Symmetric

Expectation: Variance:

Binomial distribution A salesman claims that he closes a deal 40% of the time. This month, he closed 1 out of 10 deals. How likely is it that he did 1/10 or worse given his claim?

Binomial distribution Less than 5% or 1 in 20. So, it’s unlikely that his success rate is 0.4. Note:

Binomial and normal / Gaussian distribution The normal distribution is a good approximation to the binomial distribution. (“large” n, small skew.) B(n, p) Prob. density function:

Geometric Distribution A geometric distribution is usually interpreted as number of time periods until a failure occurs. Imagine a sequence of coin flips, and the random variable X is the flip number on which the first tails occurs. The probability of a head (a success) is p.

Geometric Let’s find the probability function for the geometric distribution: etc. So, (x is a positive integer)

Geometric Notice, there is no upper limit on how large X can be Let’s check that these probabilities add to 1: Geometric series

Geometric differentiate both sides w.r.t. p: See Rosen page 158, example 17. Expectation: Variance:

Poisson distribution The Poisson distribution is typical of random variables which represent counts. Number of requests to a server in 1 hour. Number of sick days in a year for an employee.

The Poisson distribution is derived from the following underlying arrival time model: The probability of an unit arriving is uniform through time. Two items never arrive at exactly the same time. Arrivals are independent --- the arrival of one unit does not make the next unit more or less likely to arrive quickly.

Poisson distribution The probability function for the Poisson distribution with parameter  is:  is like the arrival rate --- higher means more/faster arrivals

Poisson distribution Shape Low  Med  High 