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.