Lesson 3.3: Probability Concepts and Distributions

Introduction

In this lesson, we will explore essential probability concepts and various distributions, which are foundational for understanding more complex statistical analyses in finance and investing. Our focus will be on probability rules, expected value, variance, covariance, correlation, and common distributions like the normal and lognormal distributions.

Learning Objectives

Understand and apply probability rules.
Compute expected values and variances.
Interpret covariance and correlation between variables.
Utilize the normal distribution and its standardization in problem solving.
Recognize and apply lognormal distribution in finance.

Understanding Probability Rules

Probability is a measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The fundamental rules of probability help us handle events in a systematic way.

Basic Probability Concepts

The Addition Rule: This rule is used when dealing with the probability of either of two mutually exclusive events occurring. If $ A $ and $ B $ are two mutually exclusive events, then:

$$ P(A \cup B) = P(A) + P(B) $$

The Multiplication Rule: This rule is used when calculating the probability of two independent events both occurring. If $ A $ and $ B $ are independent events, then:

$$ P(A \cap B) = P(A) \times P(B) $$

Complement Rule: The complement of an event $ A $, denoted as $ A' $, is the event that $ A $ does not occur. Thus:

$$ P(A') = 1 - P(A) $$

Example 1

Suppose a die is rolled. What is the probability of rolling a number greater than 4?

Possible outcomes: {1, 2, 3, 4, 5, 6}
Outcomes greater than 4: {5, 6}
Therefore, the probability:

$$ P(X > 4) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{6} = \frac{1}{3} $$

Expected Value

The expected value (EV) is a key concept in probability and statistics. It provides the average or mean value of a random variable based on its possible outcomes and their probabilities.

Formula for Expected Value

For a discrete random variable $ X $ with possible outcomes $ x_1, x_2, ..., x_n $ and corresponding probabilities $ P(x_1), P(x_2), ..., P(x_n) $, the expected value is:

$$ E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i) $$

Example 2

Consider a lottery game where you can win or lose. You win $100 with a probability of 0.1 and lose $10 with a probability of 0.9. The expected value calculation is:

$$ E(X) = (100 \cdot 0.1) + (-10 \cdot 0.9) $$

$$ E(X) = 10 - 9 = 1 $$

Thus, the expected value of playing this lottery game is $1.

Variance and Covariance

Variance measures the spread of a set of data points around their mean value, while covariance examines the relationship between two random variables.

Variance Formula

For a random variable $ X $ with expected value $ E(X) $, the variance $ Var(X) $ is defined as:

$$ Var(X) = E[(X - E(X))^2] = \sum_{i=1}^{n} (x_i - E(X))^2 \cdot P(x_i) $$

Example 3

Using the previous lottery game, with the outcomes being $100 and -$10, calculate the variance:

Expected Value $ E(X) = 1 $
Variance:

$$ Var(X) = (100 - 1)^2 \cdot 0.1 + (-10 - 1)^2 \cdot 0.9 $$

$$ Var(X) = (99)^2 \cdot 0.1 + (-11)^2 \cdot 0.9 $$

$$ Var(X) = 9801 \cdot 0.1 + 121 \cdot 0.9 $$

$$ Var(X) = 980.1 + 108.9 = 1089 $$

Covariance Formula

Covariance between two variables $ X $ and $ Y $ is given by:

$$ Cov(X, Y) = E[(X - E(X))(Y - E(Y))] $$

Example 4

Suppose you have two stocks, $ A $ and $ B $. Their returns for two scenarios are as follows:

Scenario 1: $ A = 10\% $, $ B = 5\% $
Scenario 2: $ A = 20\% $, $ B = 10\% $
Probabilities: $ P(Scenario 1) = 0.5 $, $ P(Scenario 2) = 0.5 $

Calculate $ E(A) $ and $ E(B) $:

Expected Return for $ A $:

$$ E(A) = 0.5 \cdot 0.10 + 0.5 \cdot 0.20 = 0.15 $$

Expected Return for $ B $:

$$ E(B) = 0.5 \cdot 0.05 + 0.5 \cdot 0.10 = 0.075 $$

Correlation

Correlation quantifies the degree to which two variables move in relation to each other, with values ranging from -1 to 1. The formula is:

ho(X, Y) = $\frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ $$

Common Probability Distributions

Two important distributions in finance are the normal distribution and the lognormal distribution.

Normal Distribution

The normal distribution is characterized by its bell-shaped curve and is described by its mean $ \mu $ and variance $ \sigma^2 $. It is widely used in finance because many variables are assumed to be normally distributed.

Properties of Normal Distribution

Symmetrical around the mean.
Approximately 68% of data falls within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Standardization of Normal Distribution

To use the normal distribution efficiently, we standardize it using the Z-score, defined as:

$$ Z = \frac{X - \mu}{\sigma} $$

Example 5

Suppose the average return of a stock is 8% with a standard deviation of 4%. What is the probability that the return will be greater than 10%?

Calculate the Z-score:

$$ Z = \frac{10 - 8}{4} = 0.5 $$

Using standard normal distribution tables or calculators, find $ P(Z > 0.5) $:

This is approximately 0.3085.

Lognormal Distribution

The lognormal distribution is useful for modeling stock prices or other financial variables that cannot be negative. If a variable $ Y $ is lognormally distributed, then $ \ln(Y) $ is normally distributed. The parameters are the mean $ \mu $ and standard deviation $ \sigma $ of the variable's natural logarithm.

Example 6

Suppose the stock price of a company is modeled to be lognormally distributed with:

Mean log return: 0.03
Variance of log return: 0.04

To calculate probabilities associated with stock prices, you can use the properties of the lognormal distribution and relevant statistical techniques.

Conclusion

In this lesson, we discussed key probability concepts, including the core principles of probability, expected value, variance, covariance, and correlation. We also reviewed common distributions such as the normal and lognormal distributions that are essential in the context of finance. Mastering these concepts equips students with the tools necessary for sound financial analysis and decision-making.

Study Notes

Probability ranges between 0 and 1.
Use the addition and multiplication rules for calculating probabilities.
Expected value provides the average outcome.
Variance measures variability, while covariance indicates the relationship between variables.
The normal distribution is characterized by its bell curve and standardized via Z-scores.
Lognormal distribution is suited for non-negative values such as stock prices.