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# Chapter 12Probability Distributions and Moment Generating Functions

This chapter describes some probability distributions that are widely used in Islamic finance. These are the uniform distribution, the Bernoulli distribution, the binomial distribution, the Poisson distribution, the normal distribution, the chi-square distribution, the student's -distribution, and the distribution. We display the principle underlying each distribution and show how probability computations may be carried out using Microsoft Excel. Many distributions are approximated by a normal distribution as the sample size or degrees of freedom become large. The probability distributions covered here are not exhaustive. There are many important probability distributions used in finance that are not covered in this chapter. A probability distribution is characterized by its moments. We discuss the moment generating function (MGF), which provides an easier way to compute moments. We show how the MGF is used to derive moments of a random variable.

## Examples of Probability Distributions

This section covers the uniform distribution, the Bernoulli distribution, the binomial distribution, the Poisson distribution, the normal distribution, the chi-square distribution, the distribution, and the distribution.

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