Capital Budgeting in the Chemical Industry Review of Elementary Statistics ChE 473K, Process Design and Operations Dr. Eldridge, Fall 2001 Gerald G. McGlamery, Jr., Ph.D., P.E. Senior Staff Team Leader Global Enterprise Management System Project ExxonMobil Global Services Company Houston, Texas

Basic Concepts of Probability The Probability Density Function The Cumulative Distribution Function Some Useful Probability Distribution Functions The Central Limit Theorem A Useful Result of the Central Limit Theorem

The Probability Density Function The Probability Density Function (PDF) expresses the probability of a continuous random variable (CRV) taking on a value between any two points in the range of that variable. Probability is expressed as: where f(x) is the probability distribution function. Probability is valid for a continuous PDF on an interval only. The probability of an event at any single value of x is zero. In a qualitative sense, the PDF does show the comparative distribution of probabilities of x taking on a specific value.

The Cumulative Distribution Function The Cumulative Distribution Function (CDF) expresses the cumulative probability of a CRV taking on any value between the lower bound and a point in the range of that variable. Probability is expressed as: where g(x) is the cumulative distribution function. The following relations hold between the PDF, f(x), and the CDF, g(x):

The Normal Distribution The Normal distribution (ND) is the most common and useful of the continuous distribution functions. The ND occurs often in nature because of the Central Limit Theorem (discussed later). The ND is a two-parameter distribution described by its mean, , and its standard deviation,  . The Standardized Normal distribution has  = 0 and  = 1. The CRV is scaled as: The Standardized Normal PDF The Normal distribution is best used to model variables whose bounds are distant from the range of interest. The Standardized Normal CDF

The Log-Normal Distribution The Log-Normal distribution (LND) is the CRV resulting from transforming the ND with the exponential function, i.e. if f(x) is the ND, then: The LND is a two-parameter distribution described by its mean, , and its standard deviation,  . The LND is best used to model asymmetric variables with a bound near the range of interest. The Log-Normal PDF The Log-Normal CDF

The Chi-Square (2) Distribution The 2 distribution is the CRV resulting from summing the squares of  normal CRVs. The 2 distribution is a one-parameter distribution described by its degrees of freedom, . The 2 distribution is best used to model asymmetric variables with a bound near the range of interest. The Chi-Square PDF The Chi-Square CDF

The Triangular Distribution The triangular distribution is a simple distribution that has no real source in nature. The triangular distribution is a three-parameter distribution described by two zero-probability endpoints and a most probable point (mode). The intensity of the mode can be proved to be: The Triangular PDF The triangular distribution is best used to model variables with upper and lower bounds near the range of interest. The Triangular CDF

The Central Limit Theorem If x is distributed with mean, , and standard deviation, , then the mean, obtained from a random sample of size n will have a distribution that approaches If x is distributed with mean, , and standard deviation, , then the xi, obtained from a random sample of size n will have a distribution that approaches

A Useful Result of the Central Limit Theorem The sum of a large number of random variables will be normally distributed regardless of the distributions of the individual random variables. Measurements of macro phenomena tend to be normally distributed because they are the sum of many micro distributions.