Chapter 7 Probability Distributions, Information about the Future

1. Chapter 7Probability Distributions, Information about the Future .

2. Random Processes

3. Random Variable • A random variable is the numerical outcome of a random (non-deterministic) process. • Intuitively, any numerically measured variable that possesses an uncertain outcome is a random variable.

4. Probability Distribution • A probability distribution is a model which describes a specific kind of random process. • Specifically, a probability distribution connects a probability to each value the random variable can assume.

5. Probability Distribution Probability Models Probability models are excellent descriptors of random processes. The following is a probability distribution (a model) for the outcome of a coin toss.

6. Types of Random Variables

7. Quantitative Random Variables • Quantitative random variables are divided into two classes. • ) Discrete • ) Continuous

8. Discrete Random Variables • A discrete random variable is a random variable which has a countable number of possible outcomes. • The values that many discrete random variables assume are the counting numbers from 0 to N, where N depends upon the nature of the variable. • Example: The number of pages in a standard math textbook is a discrete random variable. Statistics

9. Continuous Random Variables • A continuous random variable is a random variable that can assume any value on a continuous segment(s) of the real number line. • Heights, weights, volumes, and time measurements are usually measured on a continuous scale. • These measurements can take on any value in some interval.

10. Discrete or Continuous? Classify the following as either a discrete random variable or a continuous random variable. 1. the speed of a train 2. the possible scores on the SAT exam 3. the number of pizzas eaten on a college campus each day 4. the daily takeoffs at Chicago’s O’Hare Airport 5. the highest temperatures in Maine and Florida tomorrow

11. Answers 1. the speed of a train • continuous random variable 2. the possible scores on the SAT exam • discrete random variable 3. the number of pizzas eaten on a college campus each day • discrete random variable 4. the daily takeoffs at Chicago’s O’Hare Airport • discrete random variable 5. the highest temperatures in Maine and Florida tomorrow • continuous random variable

12. Naming Convention • Capital letters, such as X, will be used to refer to the random variable. example: X = number of cows in Texas • Small letters, such as x, will refer to a specific value of the random variable. example: x = 1,498,000 cows in Texas • Often the specific values will be subscripted x1, x2, ..., xn.

13. Describing a Discrete Random Variable • State (Describe) the variable. • List all of the possible values of the variable. • Determine the probabilities of these values.

14. Example 1 (die tossing) Random Phenomenon: Toss a die and observe the outcome of the toss. • X = ? • What are the possible values of X? • What are the probabilities of each value?

15. Value of X1 2 3 4 5 6 Probability Example 1 - Solution • Identify the Random Variable: X = outcome of toss of die • All possible Values: Integers between 1 and 6. In this instance x1 = 1, x2 = 2, ..., x6 = 6. • Probability Distribution: The outcomes of the toss of a die and their probabilities are given in the table. The probabilities are deduced using the classical method and the assumption of a fair die.

16. Example 2 Random Phenomenon: The head nurse of the pediatric division of the Sisters of Mercy Hospital is trying to determine the capacity requirement for the nursery. She realizes that the number of babies born at the hospital each day is a random variable. And, she will have to develop a description of the randomness in order to develop her plan. • X = ? • What are the possible values of X? • What are the probabilities of each value? • Not all discrete random variables have easily definable probability distributions.

17. Example 2 - Solution • Identify the Random Variable: X = # of babies born at hospital each day • Range of All possible Values: Integers between 0 and some large positive number. • Probability Distribution: Unknown, but could be estimated using the relative frequency idea in conjunction with historical data on hospital births.

18. Discrete Probability Distributions

19. Discrete Probability Distributions • The random variable concept is so general, that it is not very useful by itself. • What would be useful is to determine what numerical values the random variable could assume and assess the probabilities of each of these values.

20. Discrete Probability Distributions • A discrete probability distribution consists of (a list of) all possible values of the random variable with their associated probabilities. • The association of the possible values of a random variable with their respective probabilities can be expressed in three different forms: in a table, in a graph, and in an equation.

21. Characteristics Discrete probability distributions always have two characteristics: 1. The sum of all of the probabilities must equal 1. • The probability of any value must be between 0 and 1, inclusively. [Relative frequencies also share these properties]

22. Example 3 (Daily Sales) K. J. Johnson is a computer salesperson. During the last year he has kept records on his computer sales. He recognizes that his daily sales constitute a random process and wishes to determine the probability distribution for daily sales. The random variable is X = number of computers sold each day.

23. Example 3 - Solution The probabilities for this random variable are computed in the table based upon 200 days of sales data obtained from Mr. Johnson’s records using the relative frequency concept. The probability that Mr. Johnson will sell at least 2 computers each day is calculated as follows: P(X 2) = P(X=2) + P(X=3) + P(X=4) = .3 + .2 + .2 = .7. The probability that Mr. Johnson will sell at most 2 computers each day is calculated as follows: P(X 2) = P(X=0) + P(X=1) + P(X=2) = .2 + .1 + .3 = .6.

24. Example 4, Is this a prob. Dist’n? • Tell whether or not the following distribution is a probability distribution. • If the distribution is not a probability distribution, give the characteristic which is not satisfied by the distribution.

25. Example 4 - Solution Yes. All probabilities are between 0 and 1, and the sum of the probabilities is 1.

26. Example 5 , Is this a prob. Dist’n? • Tell whether or not the following distribution is a probability distribution. • If the distribution is not a probability distribution, give the characteristic which is not satisfied by the distribution.

27. Example 5 - Solution No. The sum of the probabilities is greater than one.

28. Example 6, Is this a prob. Dist’n? • Tell whether or not the following distribution is a probability distribution. • If the distribution is not a probability distribution, give the characteristic which is not satisfied by the distribution.

29. Example 6 - Solution No. You can't have negative probabilities.

30. Example 7, Is this a prob. Dist’n? • Tell whether or not the following distribution is a probability distribution. P(X=x) = , for x = 1, 2, 3, 4, 5 • If the distribution is not a probability distribution, give the characteristic which is not satisfied by the distribution.

31. Example 7 - Solution No. See table. The sum of the probabilities is 15/16 which is less than one. P(X)=x/16 for x=1 to 5 only is NOT a probability distribution.

32. Expected Value E(X) of a random variable X

33. Importance of E(X) • One of the most important concepts in the analysis of random phenomena is the notion of expected value. • Expected value is important because it is a summary statistic for a probability distribution. • It can also be used as a criteria for comparing alternative decisions in the presence of uncertainty.

34. What is Expected Value? • Conceptually, expected value is closely allied with the notion of mean or average. • The expected value is a weighted average, in which each possible value of the random variable is weighted by its probability. • Definition: • The expected value of a random variable X is the mean of the random variable X. It is denoted by E(X) and is given by computing the following expression: • = E(X) =  x* P(X=x) =  x* P(x)

35. Digression on Weighted Averages • Weighted average of any measurement (say prices Pt) is always (t wtPt )/( t wt) • weighted averages are ubiquitous. Dow Jones Industrial average is a weighted average; • see • http://www.indexarb.com/indexComponentWtsDJ.html • S&P 500 index is similar with weights available at • http://www.indexarb.com/indexComponentWtsSP500.html

36. Average Value • The expected value of a random variable should be very close to the average value of a large number of observations from the random process. • The larger the number of observations collected the more likely the expected value will be close to the average of the observations. • For discrete random variables the expected value is rarely one of the possible outcomes of the random variable.

37. E(X) for Daily Sales The expected value of the probability distribution given in Example 3 (daily Sales) is computed in the table. In the long run, data coming from a random process with this distribution should average about 2.1.

38. Using Expected Values to Compare Alternatives Two Investment Opportunities • By calculating the expected values of the two alternatives the information in each distribution is condensed to a single point. • This point characterizes the “center” of the random process and facilitates comparison. • In the long run, option B would be $500 more profitable. • But on any one investment in option B, you may lose as much as $3000 or make as much as $4000.

39. Symbols • The expected value, E(X), is the “center point” for the random process. • The symbol mx is often used to represent E(X). mx = E(X)

40. Variance and Standard Deviation of a Discrete Random Variable

41. Variance of a Discrete Random Variable • The expected value of a distribution measures only one dimension of the random variable (its central value). • To gauge the variability of a random variable we need another measure similar to the variance measure previously constructed but one which accounts for the difference in probabilities of the variable. • The variance of a discrete random variable X is given by • The larger the variance the more variability in the outcomes.

42. Standard Deviation as a measure of risk • The standard deviation is computed by taking the square root of the variance. • In the “Investment Opportunity” problem the variance and standard deviation are as follows. Option A V(X) = 3,090,000 = 1,757.84 Option B V(X) = 6,640,000 = 2,576.82 • The larger deviation reflects greater variability in profits and increased risk.

43. Sharpe Ratio • Risk adjusted returns are compared by computing the ratio • Average return / std. Dev of returns • Option A: 900/1,757.84 =0.5199199 • Option B: 1400/ 2,576.82 =0.5433053 Clearly Option B is slightly superior.

44. Example 9 Find the expected value, the variance, and the standard deviation for a random variable with the following probability distribution.

45. Example 9 - Solution

46. Probability Distributions and their Functions

47. Where do probability distributions come from? • In previous examples the distribution is already given. • In the “real world” there will be very few instances in which the probability distribution will be conveniently available. • Probabilities will have to be determined using (i) classical, (ii) relative frequency, or (iii) subjective probability. • Probability distributions can be constructed from relative frequency distributions( depicted in histograms)

48. Probability Distribution Functions (p.d.f.) • Four well known discrete distributions are: the discrete uniform, binomial, Poisson, and hypergeometric. • Each of the discrete distributions possesses a probability distribution function. • These math functions assign probabilities to each value of the random variable.

49. Discrete Probability Distribution Function Example of discrete p.d.f.: P(X=x)=1/4, if x=1,2,3,4 P(X=x)=0, otherwise This pdf does assign some value to each possible discrete number which can be the value of X. All probability values need not be positive. They can be zero!

50. Determining Probabilities for a Specific Value (just plug into the formula) To determine the probability for a specific value, use the value as the argument to the function. Pdf is P(X=x)=x2/30. [sum is unity] To determine the probability that X = 3, P(X=3) = . To determine the probability that X = 4, P(X=4) = .