1. Handout on Statistics Summary for Financial Analysis: Random Variables, Probability and Probability Distributions, Measures of Central Tendency, Dispersion, and Association
1.1 Overview of Descriptive Statistics 1.2 Random Variables 1.3 Probability and Probability Distributions 1.4 Measures of Central Tendency 1.5 Measures of Dispersion 1.6 Measures of Association

2. 1.1 Overview of Descriptive Statistics
Descriptive Statistics provides the characteristics of the probability distribution of a random variable
Useful for understanding the characteristics of the data
E.g., Rate of return: S&P 500
Mean is (μ) or E(k) is 12%
Variance (σ2) is 484%
Standard deviation (σ) is 22%
Coefficient of variation is σ / E(k) or 1.83
Skewness (Sk)assumed 0 for normal distribution, actually positively skewed distribution
Kurtosis (K) assumed 0 for normal, actually is thick-tail distributed
Jarque Bera (JB) {test of SK and K compared to normal}
E.g., Rate of return: US Treasury Bill (see lecture)

3. 1.2 Random Variables
Random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon.
Two types of random variables, discrete and continuous.
A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4, Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete.

4. 1.3 Probability and Probability Distributions
Probability distribution assigns chances (with a mathematical function) to each value of a random variable, or,
A function that assigns probabilities of occurrence to values of a random variable.
E.g. Gaussian or “normal” distribution

5. Probability Distributions
For example:
Used in Monte Carlo risk analysis to assign chances to differing possible outcomes, e.g., future CF’s and valuations for a new venture or project

6. Properties of Distributions
Usually no more than 4 parameters describe the distribution:
mean (location) { E(x) or μ}
variance (dispersion) {σ2}
skewness (symmetry){Sk}
kurtosis (thickness of tails) {K or Ku}

7. Normal Distribution Mean is the center
Variance describes the dispersion & shape
Symmetric; and has no skewness (Sk= 0)
Has no kurtosis or thick tails (K= 0)
Range is negative to positive infinity

8. Normal Distribution Function

9. Normal Distribution

10. Log-Normal Distribution
Is asymmetric
Origin is at 0
Describes data that cannot be negative:
revenues
sales
interest rates
Range is 0 to + infinity

11. Log-Normal Distribution

12. Uniform Distribution Every value has an equal chance of occurring
Has finite end-points
Has no skewness (symmetric)
Range: any two values A and B

13. Uniform Distribution

14. Binomial Distribution
Two possible outcomes with fixed probabilities that sum to 1
Range: 0 to N

15. Binomial Distribution

16. Other Distributions There are hundreds of probability distributions.
Some analysts are not concerned about probability distribution choice; too much risk in analysis makes such a choice moot
If you can reasonably choose the distribution, then do so, otherwise choose the normal.
We generally use the normal, student’s T, F distributions with regression analysis .

17. Tukey’s Boxes
Tukey’s (1977, Exploratory Data Analysis) box plots of the probability distribution of the data of a variable is a simple way to visualize the distribution of a variable
Tukey (1977) box plots show:
the 25th and 75th percentiles (the box height),
the 10th and 90th percentiles (the whiskers),
the median (the line inside the box), and,
and the dispersion of the outlying data.
The box plots should be viewed as looking down on the distributions of the betas.
See Michelfelder and Theodossiou (2013)

18. 1.4 Measures of Central Tendency
“Centers” of a probability distribution
Expected Value (e.g. Mean is a special case)
Median
Mode
Population v. sample means

19. 12.5 Measures of Dispersion
How Wide or “Squeezed” a Probability Distribution Is
Range (Maximum - Minimum)
Variance
Standard Deviation
Co-variance is not a measure of dispersion
Population v. Sample Variance and Standard Deviation

20. 1.6 Measures of Association
Relation between Random Variables
Does not estimate causation; some associations are spurious
Co-variance
Correlation Coefficient
Regression Coefficient