MGMT 201: Statistics

Discrete Probability Distributions (ASW Chapter 5)

What do we mean when we say discrete random variable? Mathematically, a discrete random variable is one for which we can devise some method for counting the possible outcomes. Note that this does not preclude having an infinite number of outcomes, as long as we have some way to count them.

Conditions

  • Let f(x) be the probability that sample point x will occur. Then,
  • f(x)  0
  • f(x) = 1
  • These are necessary conditions for a discrete probability distribution.

Attributes

  • variance = 2 = f(x)(x-)2
  • mean or expected value =  = xf(x)
  • Consider our tree from the last chapter.
  • Notice that we drew the tree in accordance with the actual chain of events. First we develop the product, second we market it, and third we sell it. This is okay, but is not the best representation because we must commit to the marketing strategy up front.

  • Now, suppose that you have assigned the following probabilities:
  • P(high cost) = 0.25
  • P(low cost) = 0.75
  • P(strong response) = 15%
  • P(mediocre response) = 60%
  • P(weak response) = 25%
  • What marketing strategy should we use?
  • Recall that we must commit to the marketing strategy now (without knowing the cost or market response).
  • Assuming that the cost and response are independent, we can assign probabilities to each outcome as follows:

PROBABILTIIES

/ Strong / Mediocre / Weak
High Cost / 0.250.15 = 0.0375 / 0.250.60 = 0.15 / 0.250.25 = 0.0625
Low Cost / 0.750.15 =0.1125 / 0.750.60 = 0.45 / 0.750.25 = 0.1875
  • Now we can address the problem by calculating our expected profits given that we choose to market to everyone, etc.

Marketing Plan

/

Expected Profits

Everyone / 0.0375$0.6 + 0.15(-$0.1) + 0.0625(-$0.5)
+ 0.1125$0.9 + 0.45$0.3 +0.1875(-$0.1) = $0.231M
Targeted / 0.0375$1.1 + 0.15$0.2 + 0.0625(-$0.1)
+ 0.1125$1.2 + 0.45$0.3 +0.1875$0.0 = $0.335M
Licensing / 0.0375$0.3 + 0.15$0.2 + 0.0625$0.1
+ 0.1125$0.5 + 0.45$0.4 +0.1875$0.2 = $0.321M
  • So, we should commit to the targeted marketing plan.

Types of Distributions

  • Discrete Uniform: Suppose we have n equally likely outcomes (e.g., rolling a die).
  • f(x) = 1/n
  • Binomial: Suppose we have a binomial situation that is repeated and we count the number of times one outcome occurs (e.g., will a customer purchase your product?).
  • Here, p is the probability of “success” for each trial and x is the number of successes out of n independent trials. p is called a parameter. By definition, a parameter is a numerical description of a population.
  • The second part of the formula (starting with px) gives us the probability of getting x successes and n-x failures.
  • The first part gives us the number of ways that can occur.
  •  = np
  • 2 = np(1-p)
  • Note that tables are available that tabulate the binomial distribution (see Table 5 in Appendix B).
  • Excel function: BINOMDIST
  • example: Consider the baseball World Series. Two teams play a best of seven series. The team that wins four games wins the series. Suppose that one team is favored and will win any given game with probability 0.6. What is the probability that the favored team will win the World Series?
  • We are interested in the outcomes for which the favored team wins at least four games. Said differently, we are interested in f(4)+f(5)+f(6)+f(7). Notice that although the series stops after one team wins four games, we can treat the unplayed games as if they would be played.
  • The problem is much more difficult when the probability varies based on where the game is played. For example, “home” teams win about 60% of the time. In a World Series, one team is designated as the home team for four games while the other is the home team for three games. If home teams indeed win 60% of the time, what is the probability that the team playing four times at home will win?
  • Let fh(x) be the distribution when the team is playing at home and fa(x) be the distribution when playing away. We will assume that game outcomes are independent.
  • For ease of exposition, call the team playing four games at home the “favorite”.
  • P(favorite wins) = fh(4) + fh(3)(fa(1)+fa(2)+fa(3)) + fh(2)(fa(2)+fa(3)) + fh(1)fa(3)
  • So, P(favorite wins) = 0.1296
    +0.3456(0.432+0.288+0.064)
    +0.3456(0.288+0.064)
    +0.15360.064
    =0.532
  • So, the team playing four games at home will win 53.2% of the time.
  • Poisson: Suppose we are interested in how many times something will occur over an interval of time (e.g., how many people will log into your web site over a given period?).
  • Here,  is the expected number of “arrivals” during the period.
  • f(x) gives the probability that exactly x people will arrive during the period.
  • See table 7 in Appendix B for Poisson values.
  • Excel function: POISSON
  • example: Suppose that a company gets, on average, 8 calls per hour to its customer service department. Management does not want customer to be put on hold very often, so it is concerned about the likelihood of getting a large number of calls during a given hour. Currently, the department can handle 12 calls per hour without putting customers on hold. Management believes it is acceptable to put people on hold during 10% of the hours. Should the customer service department hire new personnel?
  • We are interested in the probability that the company will receive 13 or more calls in a given hour.
  • P(overload) = f(13)+f(14)+f(15)+f(16)+….
  • This is an infinite series, so we are ahead to use f(x) = 1 and write
    P(overload) = 1 – (f(0) +f(1)+…+f(11)+f(12))
  • Using , the specific probabilities are as follows.

x / f(x) / x / f(x)
0 / 0.0003 / 7 / 0.1396
1 / 0.0027 / 8 / 0.1396
2 / 0.0107 / 9 / 0.1241
3 / 0.0286 / 10 / 0.0933
4 / 0.0573 / 11 / 0.0722
5 / 0.0916 / 12 / 0.0481
6 / 0.1221 / Sum / 0.9362
  • So, the probability of getting more than 12 calls in a given hour is 1-0.9362 = 0.0638. This is less than the 10% tolerance, so the department does not need to hire additional personnel.
  • Approximating the Binomial with the Poisson
  • For large n and small p, the Poisson is a good approximation of the Binomial. This is useful because the binomial can be difficult to implement for large n.
  • A good rule of thumb is that the Poisson is a good approximation when p 0.05 and n 20.
  • To use the Poisson, simply set  = np.
  • example: Suppose we have produced a lot of 100 units. We have a failure rate of 2% on average. What is the probability that the lot has 4 or less bad units?
  • We are interested in f(0)+f(1)+f(2)+f(3)+f(4).
  • The binomial model gives

x / f(x)
0 / 0.133
1 / 0.271
2 / 0.273
3 / 0.182
4 / 0.090
Total
/ 0.949
  • Using the Poisson approximation, we set =1000.02 = 2 and calculate f(x). This gives

x / f(x)
0 / 0.135
1 / 0.271
2 / 0.271
3 / 0.180
4 / 0.090
Total
/ 0.947
  • Note that Excel allows you to calculate the cumulative distributions without calculating each step.