Statistics concerns itself with distributions and properties of random variables. A random variable can be thought of as a drawing from a distribution whose outcome prior to the draw is uncertain, or stochastic. The outcome of a roll of a pair of dice, the next song that will be played on the radio, what the Fed will do in next Tuesday’s Open Market Committee meeting, next month’s peso/USD exchange rate, and next year’s return on the S&P500 index are all random variables, because the outcome is uncertain prior to the occurrence of the event. Although, of course, we cannot predict the outcome of random variables, we can form expectations of them. And forming expectations of random financial variables is at the very heart of finance. We will be developing the properties of these variables as we move through this chapter. First, we need to distinguish between discrete and continuous random variables.
A discrete random variable can take on a countable number of values (that is, finite). Discrete random variables include the roll of dice in a craps game, roulette outcomes and lottery picks and. For a roll of a single die, the ultimate outcome is unknown prior to the event taking place, but we know that there are six possible outcomes: either 1, 2, 3, 4, 5 or 6 will be rolled. There are no other possibilities. We have covered them all. One and only one of these outcomes will ensue. These facts are required in all of the distribution of outcomes. The set of possible outcomes is called the event space, or states of nature. The set of outcomes must be mutually exclusive and collectively exhaustive. When we assign a probability to each possible state of nature, we can then form expectations of the random variable.
A continuous random variable can take on any value in a given range, which may be (-∞, +∞). Examples of continuous outcomes would be those variables that have an uncountable (or infinite) number of possible outcomes. These would include stock returns. You could imagine a return of –10%, -9.9%, -9.99%, - 9.999%, …, 1.45%, 46%, … and any value, actually. With leverage you could even go beyond –100%. Tomorrow’s temperature is a random variable that will be drawn from a random distribution of possible temperatures. When senior manager Gordon ponders the future value of his Lucent stock options, he might use a continuous distribution.
Rules governing probabilities associated with distributions
1. All probabilities must be greater or equal to zero and less than or equal to one.
2. The sum of all probabilities (discrete distribution) or integral of probabilities (continuous distribution) must equal one.
3. The set of outcomes must be mutually exclusive and collectively exhaustive.
No matter the form of the random variable, we need to think about what type of distribution the outcomes will have. For the discrete variable, hopefully we can enumerate all outcomes and assign expected probabilities to them. For example, assuming we have a fair die, there is an equal probability of each outcome {1,2,3,4,5,6} occurring. Since there are six possible outcomes, there is a one in six probability of each outcome. It gets a little more complicated when we talk about a continuous random variable. We might do some scenario analysis and estimate that the probability of losing 10% on our portfolio is 23.1%. But it is unlikely that the ultimate return will be exactly what we expect. Instead, we can calculate the probability that the return will fall in some range about the expected outcome. To do this, we need to have a good idea of the nature of the distribution of returns.
Univariate distribution functions
Each random variable is a draw from some distribution, even if the distribution is unknown. The draws can be “without replacement” or “with replacement.” The roll of a die or a return on a stock is sampled with
to be “independent.” The flip of a coin, the roll of a die or a return on a stock should not depend on what happened before. Distributions of outcomes of independent random variables should exhibit no pattern or memory.
There are other distributions where successive outcomes depend on what happened before, such as random walks and Markov chains. In statistics, you often consider processes that have no memory: the value of the next (unforeseeable) observation is independent of past history; it depends only on the current level of the variable. So, if stock prices are Markovian, this means that prior history of the stock’s price process does not help predict future values. The best estimator of the next observation is the current observation. Random walks are examples of Markov chains, as are outcomes from roulette wheels, craps and other (fair) games of chance. Some lottery scheme vendors claim that future lottery picks can be guaranteed by studying the past numbers drawn. They look for patterns in the data, in a way similar to technical traders.
Discrete density function
If we let the symbol X represent the random variable “outcome of a roll of dice,” then the event set of the random variable X is the set x ={1,2,3,4,5,6}. Each outcome of this set is equally probable. There are six total outcomes, so the probability distribution associated with the outcome set is {1/6,1/6,1/6,1/6,1/6,1/6}. A univariate distribution function is a distribution function associated with a single random variable, such as this example of the roll of a die. Another term for this type of probability distribution is discrete
density function. This particular example is called a uniform density since all outcomes are equally
probable: if we graphed them, we would just have a horizontal line. Note that the area under this curve is one, as required: area = base*height = 6*1/6 = 1.
Another common example is the tossing of a fair coin. Each toss can have one of two outcomes, either tails (T) or heads (H). Like the roll of a die, a successive toss is independent of the prior result. Each outcome has a probability of 1/2 of occurring.
Multivariable distribution functions
It is very interesting to think about what happens if we combine two (or more) independent random variables. What will the resulting probability distribution and outcome space look like? The resulting distribution of outcomes of such experiments is called a joint probability distribution. A familiar example is the game of craps, in which two die are rolled simultaneously. Each die has its own distribution of outcomes and probabilities. We already found that the set of outcomes is a uniform distribution when a single die is rolled. But what happens when we roll two simultaneously? First of all, notice that we can’t get a result of “1” anymore. However, we can get a “12,” which was impossible before. The possible outcomes in this case are the sum of each individual outcome,
But does the distribution still look uniform? That is, does each outcome {2,3,4,5,6,7,8,9,10,11,12} have equal probability of occurring? The answer to this question is crucial to gambling success.
To begin to figure this out, I make a table of all of the possible outcomes:
Die 1 is … 1 2 3 4 5 6
åå
= = + = 6 1 6 1 i j j i x3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
Now we tally up the results.
Result Number of
Ways to Occur
Set of Outcomes Leading to Result Probability of Occurring 2 1 {1,1} 1/36 3 2 (1,2},{2,1} 2/36 4 3 {1,3},{2,2},{3,1} 3/36 5 4 {1,4},{2,3},{3,2},{4,1} 4/36 6 5 {1,5},{2,4},{3,3},{4,2},{5,1} 5/36 7 6 {1,6},{2,5},{3,4},{4,3},{5,2},{6,1} 6/36 8 5 {2,6},{3,5},{4,4},{5,3},{6,2} 5/36 9 4 {3,6},{4,5},{5,4},{6,3} 4/36 10 3 {4,6},{5,5},{6,4} 3/36 11 2 {5,6},{6,5} 2/36 12 1 {6,6} 1/36
(Notice that the strategies “draw a figure,” “look for a pattern,” “enumerate all cases,” and even “exploit symmetry” could have been used to solve this.) What are the probabilities of each outcome? To find out, we first sum up the number of possible outcomes: this is the sum of numbers in the second column, or 36. The probability of each individual result occurring is just the number of ways each can occur, divided by
the total number of outcomes -- this is a general result. And you will see that if you check this, the
probabilities sum to one. The probabilities for our game are shown in the last column above. The probability of “1/36” for rolling a 12, for example, means that there is only one way to roll a 12, out of 36 possible outcomes. So the next time you see someone buying a 12 in Vegas, you will know that this is a low probability bet. The number having the highest frequency (probability of occurring) is 7. You can see that the probability distribution is definitely not uniform. If you graph it, it will look like the following: