2.5 Common Probability Distributions - 2.5 Common Probability Distributions
Key Concepts
Probability distributions are mathematical functions that describe the likelihood of different possible outcomes in an experiment or survey. Understanding these distributions is crucial for making informed decisions in finance, particularly when dealing with uncertain events. The most common probability distributions include:
- Normal Distribution
- Binomial Distribution
- Poisson Distribution
- Uniform Distribution
- Exponential Distribution
Normal Distribution
The Normal Distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. It is widely used in finance to model variables such as stock returns, interest rates, and economic indicators. The distribution is characterized by its mean (μ) and standard deviation (σ).
Example: The daily returns of a stock over a year might follow a Normal Distribution. If the mean daily return is 0.1% and the standard deviation is 1.5%, we can use this distribution to estimate the probability of the stock having a daily return between -1% and 1%.
Binomial Distribution
The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is commonly used in finance to model events such as the number of times a stock will increase in value over a certain period.
Example: Suppose a stock has a 60% chance of increasing in value each day. If you want to know the probability that the stock will increase in value exactly 3 times out of 5 days, you can use the Binomial Distribution. The formula for the probability of k successes in n trials is:
P(X = k) = C(n, k) * pk * (1 - p)n-k
Where C(n, k) is the binomial coefficient.
Poisson Distribution
The Poisson Distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is often used in finance to model the frequency of certain events, such as the number of defaults in a portfolio.
Example: If a portfolio manager expects an average of 2 defaults per year, the Poisson Distribution can be used to estimate the probability of exactly 3 defaults occurring in a year. The formula for the probability of k events is:
P(X = k) = (λk * e-λ) / k!
Where λ is the average number of events and e is the base of the natural logarithm.
Uniform Distribution
The Uniform Distribution is a continuous probability distribution where all outcomes are equally likely. It is often used in finance to model situations where the outcome is equally likely to be any value within a given range, such as the price of a commodity over a specific period.
Example: If the price of a commodity is equally likely to be anywhere between $50 and $100, the Uniform Distribution can be used to estimate the probability that the price will be between $70 and $80. The probability density function is constant over the interval [a, b], where a is the minimum value and b is the maximum value.
Exponential Distribution
The Exponential Distribution is a continuous probability distribution that describes the time between events in a Poisson process, i.e., a process in which events occur continuously and independently at a constant average rate. It is often used in finance to model the time between transactions or the time until a default occurs.
Example: If the average time between transactions in a portfolio is 3 days, the Exponential Distribution can be used to estimate the probability that the next transaction will occur within 2 days. The probability density function is given by:
f(x) = λ * e-λx
Where λ is the rate parameter.