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## Chapter 4

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**Chapter 4**The curve described by a simple molecule of air vapor is regulated in a Manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance.—Marquis de LaPlace Probability Distributions and Expected Value**Random Variables andProbability Distributions**A random variable is a numerical quantity whose value is determined by chance. Number of arriving customers in an hour Rate of return on a new portfolio Your next year’s grade-point average A probability distribution may provide: A listing of possible values and probabilities. A formula for computing probabilities. A graph relating probabilities to values.**Some Probability Distributions**• Number of customers waiting for haircuts: • Number of heads in 5 coin tosses:**Expected Value**• The expected value of a random variable is the probability-weighted average. The expected number waiting is 1.38.**Meaning of Expected Value**• For repeatable random experiments, the expected value is the long-run average. • 3.5 dots for top of die cube. • 2.5 heads from 5 tosses. • 1.38 customers waiting for a haircut. • For non-repeatable experiments there is no long run. Probabilities are subjective and express “conviction” the value will occur. • A portfolio’s expected return of 13.58% is the “average conviction” for its return.**Variance of a Random Variable**• The variance of a random variable is the probability-weighted average of squared deviations from the expected value. The variance is .9995 customers squared.**Importance of Variance**• It summarizes amount of variability in the random variable. • A companion measure is the standard deviation (square root of the variance). • Units are same as the random variable itself. • In finance, variability expresses risk. • When two stocks have the same expected return, the one with the smaller variance is less risky.**Finding theProbability Distribution**• There are four ways to establish a probability distribution: • Deduction (apply probability laws & concepts) • Sum of showing dots from two rolled dice • Historical frequencies (extrapolate from past) • Fire insurance claim size • Judgment (out of “thin air” from interview) • Sales of a new product • Games played in next World Series • Assumed pattern (supported by theory) • Mean contents weight (normal distribution)**The Binomial Distribution**• Applies to a large class of random variables generated by Bernoulli processes: • Having a series of trials of like form. • Two complementary outcomes per trial: • success v. failure head v. tail defect v. satis. • Constant trial success probability. • Independent trial outcomes. • Binomial Formula:**Computing Binomial Probabilities**• The number of trials is n, the trial success probability is P, and R isthe number of successes. • R=3 defectives from n = 5 items when P = .05: • R=7 correct, n=10 random answers, P=.6 true:**About Binomial Formula**• Factorials (n!) account for possibilities: • 5! = 5×4 ×3 ×2 ×1 = 120 0! = 1 1! = 1 • Number of combinations:n!/r!(n-r)! • Pr(1-P)n-r is the probability for anyparticular r-success outcome. They are identical for same r. • Binomial formula is the product of the above. It utilizes the addition law to find the probability for any one of the several R = r possible outcomes. • Messy hand computations. Values are tabled in back of book and can be obtained on computer using Excel.**The Normal Distribution**• The normal distribution is characterized by its bell curve.**The Normal Distribution**• It has been found to give good fits to a variety of phenomena, such as physical dimensions. It is fundamental to statistics, representing levels of the sample mean. • It is totally specified by two parameters: • The mean m • The standard deviation s • Probabilities are obtained for ranges: • Correspond to areas under the normal curve. • Defined by distances from center (mean).**Finding Areas Under Normal Curve**• The area depends on the distance z in (units of s) separating a point x from m:**Poisson Distribution**• The Poisson distribution gives probabilities for number of events over time or space. • It has one parameter l, the mean rate. • Poisson probabilities are calculated from: • Cars arrive in a minute when l = 4/min.:**Poisson Process andExponential Distribution**• The Poisson process (random events over time) involves a second distribution. • The exponential distribution gives probabilities for the time between events. • The exponential cumulative probability distribution function provides the values: Pr[T<t] = 1 -e-lt • For Example, customers arrive at l = 20 per hour. Let T be the time between any 2 arrivals. Pr[T< .1 hr] = 1 - e -20(.1) = 1 - .1353=.8647**Templates and Software**• Excel Templates • Palisade Decision Tools RISKview 4.0**Excel Templates**• Expected Value, Variance, and Standard Deviation • Binomial • Normal • Exponential • Poisson**Expected Value, Variance, and Standard Deviation Template**1. Enter data in cells B4:C8. 2. If more rows are needed insert the appropriate number at any intermediate row. It is easiest not to add the new rows at the end of the table.. Figure 4-2 The Excel spreadsheet for calculating expected value, variance, and standard deviation for the price of ChipMont**Excel’s BINOMDIST Function**=BINOMDIST(r,n,P,cumulative) • r = number of successes • n = number of trials • P = success probability • cumulative = TRUE for cumulative distribution and FALSE for individual probabilities**2. If more rows are needed insert them at any intermediate**row. Copy the formulas in columns B and C down to the inserted rows and onwards to the bottom of the expanded table. Binomial Probability Distribution Excel Template 1. Enter new data in C4 and G4. Figure 4-8 (upper portion) Excel spreadsheet and graph for finding binomial probabilities for number of persons remembering aspirin ad**Binomial Probability Distribution**Figure 4-8 (lower portion) Excel spreadsheet and graph for finding binomial probabilities for number of persons remembering aspirin ad**Excel’s NORMDIST Function**=NORMDIST(t,m,s,cumulative) • t = value for which the normal probability is being calculated • m = mean • s = standard deviation • cumulative = TRUE for cumulative distribution and FALSE for individual probabilities**Normal DistributionExcel Template**2. If more rows are needed insert them at any intermediate row. Enter the data in column A. Copy the formulas in columns B and C down to the inserted rows and onwards to the bottom of the expanded table. 1. Enter new data in C3:C4. Figure 4-13 (upper portion) Spreadsheet for typesetting normal distribution**Normal Distribution**The graph here has equal time intervals so the curves are smoother and the normal frequency curve is multiplied by 50 to be able to see it better. Figure 4-13 (lower portion) Spreadsheet for typesetting normal distribution**Excel’s EXPONDIST Function**• =EXPONDIST(t, l,cumulative) • t = value for which the exponential probability is being calculated • l = mean arrival rate • s = standard deviation • cumulative = TRUE for cumulative distribution and FALSE for individual probabilities**Exponential DistributionExcel Template**2. If more rows are needed insert them at any intermediate row. Enter the data in column A. Copy the formulas in columns B and C down to the inserted rows and onwards to the bottom of the expanded table. 1. Enter new data in C3. Figure 4-16 (upper portion) Spreadsheet for exponential distribution**Exponential Distribution**Figure 4-16 (lower portion) Graph for exponential distribution**Excel’s POISSON Function**• =POISSON(X, mean,cumulative) • X = number of events during time period t • mean = lt • l = mean number of events per unit time • cumulative = TRUE for cumulative distribution and FALSE for individual probabilities**Poisson DistributionExcel Template**2. If more rows are needed insert them at any intermediate row. Enter the data in column A. Copy the formulas in columns B and C down to the inserted rows and onwards to the bottom of the expanded table. 1. Enter new data in C3:C4. Figure 4-17 (upper portion)**Poisson Distribution**Figure 4-17 (lower portion) Graph of the Poisson density function and the cumulative distribution for number of arrivals**Palisade Decision ToolsRISKview**The RISKview 4.0 software program on the CD-ROM accompanying this book provides a picture of more than 30 different distributions. A few of the more common distributions beta, binomial, chi-square, exponential, gamma, geometric, hypergeometric, normal, Poisson, triangular, and uniform.**RISKview**To start RISKview, click on the Windows Start button, select Programs, Palisade Decision Tools, then RISKview 4.0. RISKview will open and an initial screen like the one shown next will appear.**Initial RISKview Screen**1. Click on down arrow in the Dist line to show a list of the other distributions RISKview can display. 2. Enter the mean in the m line. 3. Enter the std. dev. in the s line.