11.1 discrete probability distributions

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CHAPTER 2

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Discrete Probability

Distributions

ill Microsofts stock return over the next year exceed 10%? Will the1-month London Interbank Offered Rate (LIBOR) three months

from now exceed 4%? Will Ford Motor Company default on its debtobligations sometime over the next five years? Microsofts stock returnover the next year, 1-month LIBOR three months from now, and thedefault of Ford Motor Company on its debt obligations are each vari-

ables that exhibit randomness. Hence these variables are referred to asrandom variables.1 In the chapters in Part One, we will see how proba-bility distributions are used to describe the potential outcomes of a ran-dom variable, the general properties of probability distributions, andthe different types of probability distributions.2 Random variables canbe classified as either discrete or continuous. In this chapter, our focus ison discrete probability distributions.

1 The precise mathematical definition is that a random variable is a measurablefunction from a probability space into the set of real numbers. In the following thereader will repeatedly be confronted with imprecise definitions. The authors have

intentionally chosen this way for a better general understandability and for sake ofan intuitive and illustrative description of the main concepts of probability theory.The reader already familiar with these concepts is invited to skip this and some ofthe following chapters. In order to inform about every occurrence of looseness andlack of mathematical rigor, we have furnished most imprecise definitions with afootnote giving a reference to the exact definition.2 For more detailed and/or complementary information, the reader is referred to thetextbook by Larsen and Marx (1986) or Billingsley (1995).

W


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14 PROBABILITY AND STATISTICS

BASIC CONCEPTS

An outcome for a random variable is the mutually exclusive potentialresult that can occur. A sample space is a set of all possible outcomes.An event is a subset of the sample space.3 For example, considerMicrosofts stock return over the next year. The sample space containsoutcomes ranging from 100% (all the funds invested in Microsoftsstock will be lost) to an extremely high positive return. The samplespace can be partitioned into two subsets: outcomes where the return isless than or equal to 10% and a subset where the return exceeds 10%.

Consequently, a return greater than 10% is an event since it is a subsetof the sample space. Similarly, a 1-month LIBOR three months fromnow that exceeds 4% is an event.

DISCRETE PROBABILITY DISTRIBUTIONS DEFINED

As the name indicates, a discrete random variable limits the outcomeswhere the variable can only take on discrete values. For example, con-sider the default of a corporation on its debt obligations over the nextfive years. This random variable has only two possible outcomes: defaultor nondefault. Hence, it is a discrete random variable. Consider an

option contract where, for an upfront payment (i.e., the option price) of$50,000, the buyer of the contract receives the following payment fromthe seller of the option depending on the return on the S&P 500 index:

In this case, the random variable is a discrete random variable but onthe limited number of outcomes.

The probabilistic treatment of discrete random variables is compara-tively easy: Once a probability is assigned to all different outcomes, theprobability of an arbitrary event can be calculated by simply adding the

3 Precisely, only certain subsets of the sample space are called events. In the case thatthe sample space is represented by a subinterval of the real numbers, the events con-sist of the so-called Borel sets. For all practical applications, we can think of Borelsets as containing all subsets of the sample space.

If S&P 500 return is Payment received by option buyer

Less than or equal to zero $0

Greater than zero but less than 5% $10,000

Greater than 5% but less than 10% $20,000

Greater than or equal to 10% $100,000


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Discrete Probability Distributions 15

single probabilities. Imagine that in the previous example on the S&P500 every different payment occurs with the same probability of 25%.Then the probability of losing money by having invested $50,000 to pur-chase the option is 75%, which is the sum of the probabilities of gettingeither $0, $10,000, or $20,000 back.

In the following sections we provide a short introduction to the mostimportant discrete probability distributions: Bernoulli distribution, Bino-mial distribution, and Poisson distribution. A detailed description, togetherwith an introduction to several other discrete probability distributions, canbe found, for example, in the textbook by Johnson, Kotz, and Kemp (1993).

BERNOULLI DISTRIBUTION

We will start the exposition with the Bernoulli distribution. A randomvariable X is called Bernoulli distributedwith parameter p if it has onlytwo possible outcomes, usually encoded as 1 (which might representsuccess or default) or 0 (which might represent failure or sur-vival) and if the probability for realizing 1 equalsp and the probabil-ity for 0 equals 1 p.

One classical example for a Bernoulli-distributed random variableoccurring in the field of finance is the default event of a company. We

observe a company C in a specified time interval I, e.g., January 1,2006, until December 31, 2006. We define

The parameter p in this case would be the annualized probability ofdefault of company C.

BINOMIAL DISTRIBUTION

In practical applications, we usually do not consider only one single com-pany but a whole basket C1,Cn of companies. Assuming that all these ncompanies have the same annualized probability of defaultp, this leads toa natural generalization of the Bernoulli distribution, called Binomial dis-tribution. A Binomial distributed random variable Ywith parameters nandp is obtained as the sum ofn independent4 and identically Bernoulli-distributed random variables X1,,Xn. In our example, Yrepresents the

X1 ifC defaults in I0 else

=


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16 PROBABILITY AND STATISTICS

total number of defaults occurring in the year 2006 observed for compa-nies C1,Cn. Given the two parameters, the probability of observing k,0 kn defaults can be explicitly calculated as follows:

The notation

means

Recall that the factorial of a positive integer n is denoted by n! and isequal to n(n 1)(n 2) 2 1. Exhibit 2.1 provides a graphical visu-alization of the Binomial probability distribution for several differentparameter values.

POISSON DISTRIBUTION

The last distribution that we treat in this chapter is the Poisson distribu-tion. The Poisson distribution depends upon only one parameter andcan be interpreted as an approximation to the binomial distribution. APoisson-distributed random variable is usually used to describe the ran-dom number of events occurring over a certain time interval. We usedthis previously in terms of the number of defaults. One main differencecompared to the binomial distribution is that the number of events thatmight occur is unboundedat least theoretically. The parameter indi-cates the rate of occurrence of the random events, that is, it tells us howmany events occur on average per unit of time.

The probability distribution of a Poisson-distributed random vari-able Nis described by the following equation:

, k = 0, 1, 2,

4 A definition of what independence means is provided in Chapter 5. The readermight think of independence as no-interference between the random variables.

P Y k=( ) nk

pk 1 p( )n k=

n

k

n!

n k( )!k!-------------------------

P N k=( )k

k!------ e

=


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Discrete Probability Distributions 21

EXHIBIT 2.2 (Continued)

REFERENCES

Billingsley, P. 1995. Probability and Measure: Third Edition. New York: John Wiley& Sons.

Johnson, N. L., S. Kotz, and A. W. Kemp, 1993. Univariate Discrete Distributions.Second Edition, New York: John Wiley & Sons.

Larsen, R. J. and M. L. Marx. 1986. An Introduction to Mathematical Statistics andits Applications. Englewood Cliffs, NJ: Prentice Hall.

11.1 discrete probability distributions

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