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George Mason UniversityConor Philip NelsonSource: Home > Classes > Math 400 > Final Project > ApplicationMath 400: History of MathematicsIn this section I will explain about the application of the Gamma function to aspects in Probability. First we will introduce the concept of a distribution function through a Binomial Distribution. First consider the concept of a Bernoulli Random variable. This means that given an event with one trial, there is a probability of success \(p\) and a probability of a success \(1-p\). We can now consider \(n\) events of these Bernoulli Distributions, this is known as the Binomial distribution. It's Probability Density function is of this form: $$p(i)={n\choose i}p^i(1-p)^{n-i}$$ This represents the number of successes the that occur in \(n\) trials. We will now consider that success is represented by a function. This generates different Random Variables. Consider the Exponential Random Variables: $$p_e(x)=\lambda e^{-\lambda x} ... \forall x\geq 0$$ and 0 elsewhere. This represents the amount of time until some specific even occurs. Some examples of this include, the time it takes until a war breaks out, until the next telephone call you receive was a wrong number or until the next earthquake occurs. Similarly, lets consider a function of this form: $$p_{\Gamma}=\frac{\lambda e^{-\lambda x}(\lambda x)^{\alpha -1}}{\Gamma(\alpha)}$$ This is the Gamma Distribution and it represents the amount of time one has to wait until a total of \(n\) events has occurred. Consider this final function: $$p_{\beta}(x)=\frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}... \forall x \geq 0 $$ and 0 else where. Note that \(B(a,b)\) is of the form: $$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ This the Beta Distribution, which is used to model random phenomenon where the possible values are in some finite interval \([c,d]\).
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