Quantitative Methods - Common Probability Distribution Properties

Normal Distribution
The normal distribution is a continuous probability distribution that, when graphed as a probability density, takes the form of the so-called bell-shaped curve. The bell shape results from the fact that, while the range of possible outcomes is infinite (negative infinity to positive infinity), most of the potential outcomes tend to be clustered relatively close to the distribution's mean value. Just how close they are clustered is given by the standard deviation. In other words, a normal distribution is described completely by two parameters: its mean (μ) and its standard deviation (σ).

Here are other defining characteristics of the normal distribution: it is symmetric, meaning the mean value divides the distribution in half and one side is the exact mirror image of the other -that is, skewness = 0. Symmetry also requires that mean = median = mode. Its kurtosis (measure of peakedness) is 3 and its excess kurtosis (kurtosis - 3) equals 0. Also, if given 2 or more normally distributed random variables, the linear combination must also be normally distributed.

While any normal distribution will share these defining characteristics, the mean and standard deviation will be unique to the random variable, and these differences will affect the shape of the distribution. On the following page are two normal distributions, each with the same mean, but the distribution with the dotted line has a higher standard deviation.

Univariate vs. Multivariate Distributions
A univariate distribution specifies probabilities for a single random variable, while a multivariate distribution combines the outcomes of a group of random variables and summarizes probabilities for the group. For example, a stock will have a distribution of possible return outcomes; those outcomes when summarized would be in a univariable distribution. A portfolio of 20 stocks could have return outcomes described in terms of 20 separate univariate distributions, or as one multivariate distribution.

Earlier we indicated that a normal distribution is completely described by two parameters: its mean and standard deviation. This statement is true of a univariate distribution. For models of multivariate returns, the mean and standard deviation of each variable do not completely describe the multivariate set. A third parameter is required, the correlation, or co-movement, between each pair of variables in the set. For example, if a multivariate return distribution was being assembled for a portfolio of stocks, and a number of pairs were found to be inversely related (i.e. one increases at the same time the other decreases), then we must consider the overall effect on portfolio variance. For a group of assets that are not completely positively related, there is the opportunity to reduce overall risk (variance) as a result of the interrelationships.

For a portfolio distribution with n stocks, the multivariate distribution is completely described by the n mean returns, the n standard deviations and the n*(n - 1)/2 correlations. For a 20-stock portfolio, that's 20 lists of returns, 20 lists of variances of return and 20*19/2, or 190 correlations.

Confidence Intervals

Related Articles
  1. Term

    What a Normal Distribution Means

    Normal distribution describes a symmetrical data distribution, where most of the results lie near the mean.
  2. Fundamental Analysis

    Find The Right Fit With Probability Distributions

    Discover a few of the most popular probability distributions and how to calculate them.
  3. Forex Education

    Trading With Gaussian Models Of Statistics

    The entire study of statistics originated from Gauss and allowed us to understand markets, prices and probabilities, among other applications.
  4. Investing Basics

    Using Normal Distribution Formula To Optimize Your Portfolio

    Normal or bell curve distribution can be used in portfolio theory to help portfolio managers maximize return and minimize risk.
  5. Options & Futures

    Multivariate Models: The Monte Carlo Analysis

    This decision-making tool integrates the idea that every decision has an impact on overall risk.
  6. Fundamental Analysis

    Scenario Analysis Provides Glimpse Of Portfolio Potential

    This statistical method estimates how far a stock might fall in a worst-case scenario.
  7. Active Trading Fundamentals

    Bet Smarter With The Monte Carlo Simulation

    This technique can reduce uncertainty in estimating future outcomes.
  8. Fundamental Analysis

    Quantitative Analysis Of Hedge Funds

    Hedge fund analysis requires more than just the metrics used to analyze mutual funds.
  9. Fundamental Analysis

    Explaining the Empirical Rule

    The empirical rule provides a quick estimate of the spread of data in a normal statistical distribution.
  10. Financial Advisors

    How to Navigate Taxable Mutual Fund Distributions

    It's almost time for year-end capital gains distributions for mutual funds. Here's how to monitor them and minimize their tax impact.
  1. Probability Distribution

    A statistical function that describes all the possible values ...
  2. T Distribution

    A type of probability distribution that is theoretical and resembles ...
  3. Bell Curve

    The most common type of distribution for a variable. The term ...
  4. Normal Distribution

    A probability distribution that plots all of its values in a ...
  5. Multivariate Model

    A popular statistical tool that uses multiple variables to forecast ...
  6. Mesokurtic

    A term used in a statistical context where the kurtosis of a ...
  1. How is standard deviation used to determine risk?

    Understand the basics of calculation and interpretation of standard deviation and how it is used to measure risk in the investment ... Read Answer >>
  2. What does standard deviation measure in a portfolio?

    Dig deeper into the investment uses of, and mathematical principles behind, standard deviation as a measurement of portfolio ... Read Answer >>
  3. Can I elect to NOT have income tax withheld from an IRA (NOT ROTH) distribution before ...

  4. Is my non-qualified Roth IRA distribution subject to taxes or early distribution ...

    The ordering rules must be applied to determine whether the distribution is subject to income taxes and/or the early distribution ... Read Answer >>
  5. What is the difference between standard deviation and average deviation?

    Understand the basics of standard deviation and average deviation, including how each is calculated and why standard deviation ... Read Answer >>
  6. What are the exceptions to the early distribution penalty for a non-qualified Roth ...

    The exceptions are as follows: The distribution is made on or after the date you reach age 59.5 The distribution is made ... Read Answer >>
Hot Definitions
  1. Quarter - Q1, Q2, Q3, Q4

    A three-month period on a financial calendar that acts as a basis for the reporting of earnings and the paying of dividends.
  2. Weighted Average Cost Of Capital - WACC

    Weighted average cost of capital (WACC) is a calculation of a firm's cost of capital in which each category of capital is ...
  3. Basis Point (BPS)

    A unit that is equal to 1/100th of 1%, and is used to denote the change in a financial instrument. The basis point is commonly ...
  4. Sharing Economy

    An economic model in which individuals are able to borrow or rent assets owned by someone else.
  5. Unlevered Beta

    A type of metric that compares the risk of an unlevered company to the risk of the market. The unlevered beta is the beta ...
  6. Treasury Inflation Protected Securities - TIPS

    A treasury security that is indexed to inflation in order to protect investors from the negative effects of inflation. TIPS ...
Trading Center