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Multinomial Distribution – Definition – Uses in Finance – Examples

What Is a Multinomial Distribution?

Multinomial DistributionA Multinomial distribution is the data set from a multinomial experiment. It is an extension of binomial distribution in that it has more than two possible outcomes.

A multinomial distribution is a type of probability distribution.  It is the result when calculating the outcomes of experiments involving two or more variables. A binomial distribution is a special type of multinomial distribution in which there are only two possible outcomes.  The most common are experiments where the result is true or false, heads or tails.

In finance, analysts use multinomial distributions to estimate the probability of a given set of outcomes occurring.  For example, the likelihood that a company will increase its sales for the quarter when its competition reports lower than expected earnings.

The multinomial distribution applies to experiments with the following conditions:

  • Repeated trials – The experiment consists of repeated trials instead of one attempt.  For example, rolling a dice ten times instead of just once.
  • Each trial is independent of the others. For example, if you roll two dice, the outcome of one die does not impact the outcome of the other die.
  • The probability does not change – Each outcome must be the same across each instance of the experiment. For example, if a dice has six sides, then there must be a one in six chance of each number being given on each roll.
  • A measurable result – Each trial must produce a specific outcome.  For example, a number between two and 12 if rolling two six-sided dice.


For example, suppose we conduct an experiment by rolling two dice 100 times. The goal is to calculate the probability that the experiment will produce the following results across the 100 trials:

  • An outcome will be “2” in 3% of the trials;
  • The outcome will be “6” in 14% of the trials;
  • An outcome will be “7” in 18% of the trials; and
  • The outcome will be “10” in 8% of the trials.

A multinomial distribution allows us to calculate the probability that the above combination of outcomes will occur. The same type of analysis can be performed for meaningful experiments in science, investing, finance, and other areas.

Multinomial Distribution in Finance and Investing

In the context of investing, a portfolio manager or financial analyst might use the multinomial distribution to estimate probability.  For example, a small-cap index outperforming a large-cap index 60% of the time.  Or, the large-cap index outperforming the small-cap index 40% of the time.  Or, that the indexes having the same or approximate return 10% of the time.  In this scenario, the trial might take place over a full year of trading days.  Actual market data can be used to verify the results. If the probability of this set of outcomes is sufficiently high, the investor might be tempted to make an overweight investment in the small-cap index. (Source:

Final Words

The binomial distribution is a more well-known discrete distribution.  However, the multinomial distribution provides a useful generalization to the binomial distribution. As the names of the distributions clarify, the binomial distribution makes use of the binomial coefficient which comes from the binomial theorem.  Conversely, the multinomial distribution makes use of the multinomial coefficient which comes from the multinomial theorem. Multinomial logistic regression and logistic regression are generalized linear models.  The dependent variable (outcome variable) for a multinomial logistic regression follows a multinomial distribution.  On the other hand, for the logistic regression, the dependent variable follows a binomial distribution.

Up Next: What Is the Sum of Squares?

Sum of SquaresThe Sum of squares is a statistical technique to measure how dispersed the numbers in a dataset are and their deviation from a mean data point

The Sum of squares is used in statistics for regression analysis to determine the dispersion of data points. In a regression analysis, the objective is to determine how well a data series can fit a particular function.  In turn, this provides clues to help explain how the data series was generated. Ultimately, the sum of squares is a mathematical way to find the function that best fits the data. In regression analysis, it is a way to measure variance.  That means how dispersed and spread out the numbers in a dataset are. The sum of squares gets its name from the way you calculate it.  By summing up the squared difference between an observation and the target value.

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