Discrete Probability Distributions
Definitions
Area of Opportunity: (for Poisson distribution) a continuous interval (time, length, area, volume, etc).
Binomial Distribution has the following properties:
- The experiment consists of n repeated trials.
- Each trial can result in just two possible outcomes, success or failure.
- The probability of success is the same on every trial.
- The trials are independent.
Continuous Variable: can take on any value between two specified values. Obtained by measuring.
Covariance: a measure of the direction of the linear relationship between two variables.
Discrete Variable: not continuous variable (cannot take on any value between two specified values). Obtained by counting.
Expected Value: (a.k.a. Mean, or Expectation, or Mathematical Expectation) the probability-weighted average of all possible values in a distribution.
Mean: see Expected Value.
Poisson Distribution: a discrete frequency distribution that gives the probability of a number of independent events occurring in a fixed area of opportunity.
Portfolio: a collection of investments held by an investor.
Probability Distribution Function: (for a discrete variable) a list of all possible values of a discrete variable X and the probabilities that X can take each of those possible values.
Combination: a selection of a part of a set of objects, without regard to the order in which the objects are selected.
Standard Deviation: a measure of dispersion that is equal to the square root of the variance.
Variance: the probability-weighted average of the squared differences between all possible values in a distribution and the distribution's expected value.
These are the illustrations I have used in class:
The Binomial Distribution Applet from Matt Bognar's (Department of Statistics and Actuarial Science, University of Iowa) page. Try to enter various values for n and p to see how the shape of the binomial distribution depends on these two parameters [p there is the probability of the event of interest in a single trial]. Entering a value for x will also calculate the probability of the variable being equal to that value.
- Try entering n = 100 and p = 0.5. Do you see anything peculiar about the shape of the distribution?
The Poisson Distribution Applet, also from Matt Bognar's page. Try to enter various values for λ to see how the shape of the binomial distribution depends on this parameter. Entering a value for x will also calculate the probability of the variable being equal to that value.
- Try entering λ = 100. Do you see anything peculiar about the shape of the distribution?
Notes
We do not really know whether there is anything continuous in the real world (asking a physicist about the Planck constant may be a right place to start learning about this). I suggest that the best way to think about the continuous distributions is to regard them as mathematical models that are a convenient approximation to the reality when the (possible) discrete nature of the world can be ignored (the benefit? a simpler model).
Recall the column charts and the histograms. We have gaps in a column chart to remind us that the categories are "disjoint," and no gaps in a histogram to remind us that the values in a numerical class can be anywhere within the numerical class width. Discrete variable's values are also "disjoint," so we have the gaps in a discrete variable's probability distribution graph.
While the formula for the expected value of a discrete variable may seem different from the formula for the mean of a set of values, it is the exactly same formula, just written a different way. Same goes for the variance, standard deviation, and the covariance between two variables.
A risky (uncertain) investment has two important characteristics to describe the distribution of a random return:
- Expected return measured by the mathematical expectation of the return distribution;
- Risk measured by the variance (or standard deviation) of the return distribution;
A higher expected return is desirable, while the risk is undesirable.
A portfolio offers an opportunity to create an investment with lower risk without sacrificing too much of the expected return. To achieve this we have to combine the individual assets for which the covariance is negative.
Read These
Chapter 5. Discrete Probability Distributions in the textbook:
5.1 The Probability Distribution for a Discrete Variable (pp. 186-188)
- It may be a good idea to work on your own to calculate the mean, variance, and standard deviation for the distribution in Table 5.1 (p. 186). the answers are in Table 5.3 (p. 188).
5.2 Covariance of a Probability Distribution and Its Application in Finance (pp. 189-193)
5.3 Binomial Distribution (pp. 195-201)
- You may omit "The Side-by-Side Bar Chart" section.
- Pay attention to the paragraph that argues for and against the use of the pie charts.
5.4 Poisson Distribution (pp. 202-204)
You may omit sections 5.5 Hypergeometric Distribution and 5.6 Using the Poisson Distribution to Approximate the Binomial Distribution (pp. 206-209)
Watch These
Figure 0060.050. How to Calculate the Risk and Expected Return of Portfolios . This gives an example and details.
Answer These
Calculate the expected value, variance, and standard deviation for the following distribution:
| Probability | Value |
|---|---|
| 0.06 | 200 |
| 0.11 | 240 |
| 0.11 | 280 |
| 0.22 | 320 |
| 0.12 | 360 |
| 0.20 | 400 |
| 0.18 | 440 |
Do problem 5.12 (p. 194 in the textbook).
A fair coin is tossed 10 times. Determine the probability distribution for the number of times the coin lands Heads Up out of 10 tosses; build a probability distribution graph.
Do problem 5.23 (p. 202 in the textbook).
File SeriaA.xlsx has the data for every game in Italian Seria A, season 2015-2016. I have added and highlighted two columns, Goals for the goals scored in each game and Shots for the shots made in each game. Download the file, and use the data to fill the summary table below:
- For Number of Games (actual), count the number of games in which 0 goals were scored, the number of games in which 1 goal was scored, the number of games in which 3 goals were scored, etc.
- For Number of Games (Poisson), calculate the number of games in which 0 goals would be scored, the number of games in which 1 goal would be scored, the number of games in which 3 goals would be scored, etc. if the number of goals in a game followed a Poisson distribution.
- For Number of Games (Binomial), calculate the number of games in which 0 goals would be scored, the number of games in which 1 goal would be scored, the number of games in which 3 goals would be scored, etc. if the number of goals in a game followed a Binomial distribution where the number of trials per game is the mean number of shots per game over the season and the probability of scoring a goal per shot is the mean number of goals per shot over the season.
- Finally, create a chart to compare the actual number of the games with each number of goals scored to the numbers predicted by Poisson and Binomial distributions.
| Goals in a Game | Number of Games (actual) | Number of Games (Poisson) | Number of Games (Binomial) |
|---|---|---|---|
| 0 | |||
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7 | |||
| 8 | |||
| 9 | |||
| 10 | |||
| Total | 380 |
Figure 0060.040. How to find the Expected Return and Risk. An example and details of calculating the expectation and the variance (+ standard deviation) of a random variable (return, in this case).