Lecture 4 Section 1.5 Objectives: Other Continuous Distributions Exponential Distribution Lognormal Distribution Weibull Distribution

Exponential Distribution Normal density curves are always bell-shaped and symmetric. Several useful skewed distributions are now presented. Definition. A variable x is said to have an exponential distribution with parameter λ > 0 if the density function of x is Note: Exponential density curves are right-skewed (positively skewed), have their maximum at x=0 and decreases as x increases. b. Exponential distribution has been used to model the waiting times between successive changes, for example, time between successive arrivals at a service facility, the amount of time to complete a specified task, and the 1-hr concentration of carbon monoxide in an air sample.

Example 1.12 Let x denote the response time (sec) at a certain online computer; that is, x is the time between the end of a user’s inquiry and the beginning of the system’s response to that inquiry. The value of x varies from inquiry to inquiry. Suppose x has an exponential distribution with λ = 0.2. Find the proportion of inquiries with a response time less than 5 sec. b. Find the value c that separates the largest 10% of all times from the smallest 90%.

Lognormal Distribution Definition. A variable x is said to have a lognormal distribution if ln(x) has a normal distribution with parameters μ and σ. It can be shown that the density function of x is

Example 1.18 The lognormal distribution is frequently used as a model for various material properties. The article “Reliability of Wood Joist Floor Systems with Creep” (J. of Structural Engr., 1995:946-954) suggests that the lognormal distribution with μ = .375 and σ = .25 is a plausible model for x=the modulus of elasticity (MOE, in 106 psi) of wood joist floor systems constructed from #2 grade hem-fir. Find the proportion of systems with x < 2. b. Find the proportion of systems with 1 < x < 2. c. What value c is such that only 1% of all systems have an MOE exceeding c?

Weibull Distribution Definition. A variable x has a Weibull distribution with parameters α and β if the density function of x is Note: When α = 1, the Weibull distribution reduces to the exponential distribution with λ = 1/β. b. Proportion of x values satisfying x < t =

Example 1.19 In recent years the Weibull distribution has been used to model engine emissions of various pollutants. Let x denote the amount of NOx emission (g/gal) from a certain type of four-stroke engine, and suppose that x has a Weibull distribution with α = 2 and β = 10 (suggested by information in the article “Quantification of Variability and Uncertainty in Lawn and Garden Equipment NOx and Total Hydrocarbon Emission Factors”, J. of the Air and Waste Management Assoc., 2002: 435-448). Find the proportion of engines emitting less than 10 g/gal. Find the proportion of engines emitting at most 25 g/gal. Find the value c which separates the 5% of all engines having the largest amounts of NOx emissions from the remaining 95%.