Understanding common probability distributions is crucial for investment analysis. These distributions help us model the likelihood of different outcomes for financial variables like asset returns, interest rates, and economic indicators. This knowledge is fundamental for risk management, portfolio construction, and valuation.

A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can take. In simpler terms, it tells us how likely different outcomes are. For continuous random variables, we use probability density functions (PDFs), and for discrete random variables, we use probability mass functions (PMFs).

Several probability distributions are particularly relevant in investment analysis. We'll explore some of the most common ones:

The normal distribution, often called the 'bell curve,' is a symmetric, continuous probability distribution. It's characterized by its mean ($\mu$) and standard deviation ($\sigma$). Many financial variables, especially asset returns over short periods, are often assumed to follow a normal distribution. It's fundamental for calculating risk metrics like Value at Risk (VaR).

The lognormal distribution is used for variables that cannot take negative values and tend to grow exponentially. Asset prices, for example, are often modeled using a lognormal distribution because prices cannot go below zero and can theoretically increase indefinitely. If the logarithm of a random variable is normally distributed, then the variable itself is lognormally distributed.

The Student's t-distribution is similar to the normal distribution but has 'fatter tails.' This means it assigns a higher probability to extreme values than the normal distribution. It's often used when dealing with small sample sizes or when the underlying data exhibits more extreme outliers than a normal distribution would suggest. The shape of the t-distribution depends on its 'degrees of freedom'.

The Bernoulli distribution is a discrete distribution for a random variable that can take only two possible values: 0 (failure) or 1 (success). It's often used to model binary outcomes, such as whether a stock price will go up or down on a given day, or whether a bond will default. The distribution is defined by a single parameter, $p$, which is the probability of success.

The binomial distribution is a discrete distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. For example, it could model the number of successful trades out of 10 attempts, or the number of companies in a sector that report positive earnings in a quarter.

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's useful for modeling rare events, such as the number of defaults in a portfolio over a year, or the number of trading halts in a day.

Understanding these distributions allows analysts to:

• Quantify Risk: Estimate the probability of adverse events (e.g., large losses) using metrics like VaR, which often relies on normal or t-distributions.
• Model Asset Prices: Use lognormal distributions to forecast potential future asset prices.
• Price Derivatives: Many option pricing models, like the Black-Scholes model, assume underlying asset prices follow a lognormal distribution.
• Forecast Event Probabilities: Use Bernoulli, binomial, or Poisson distributions to model the likelihood of specific events like defaults or successful investments.

Mastering common probability distributions is a cornerstone of quantitative finance. Each distribution has unique properties that make it suitable for modeling different financial phenomena. By understanding their characteristics and applications, you can build more robust financial models and make more informed investment decisions.