5

I am not going to wade through the initial queueing model, so I'll just take your word for it that you want to find the expected value of $e^{vS}$ where $S$ is an exponential random variable. This being the case, what you are looking for here is the moment generating function of an exponential random variable. From what you say in your comments, it appears ...


4

It is best to look at the distribution of waiting times for a particular provider. My first thought would be that if the process is anything like a queueing If process that the distribution should be nearly exponential. So I would check to see if the sample mean and standard deviation are approximately equal. If so, I would look to see if an empirical CDF (...


2

Yes it's possible. Suppose that $X\sim \mathrm{Gamma}(\alpha, \beta)$ where $\alpha$ is the shape parameter and $\beta$ the scale parameter. Define the incomplete gamma function as $$ \Gamma(a, z)=\int_{z}^{\infty}t^{a-1}\mathrm{e}^{-t}\;\mathrm{d}t $$ and the generalized incomplete gamma function as $$ \Gamma(a, z_0, z_1)=\int_{z_0}^{z_1}t^{a-1}\mathrm{e}...


2

If $X$ and $Y$ are both standard normal then $X^2 + Y^2 \sim \chi^2_2 = \mathcal{E}(1/2)$ and the angle of $(X,Y)$ in the plane whose sinus is given by $\frac{Y}{\sqrt{X^2 + Y^2}}$ is $\mathcal{U}_{[-\pi,\pi]}$. Thus let $\theta \sim \mathcal{U}_{[-\pi,\pi]}$ and $Z \sim \mathcal{E}(1/2)$, then $$ Y = \sqrt{Z} \text{sin}(\theta) \sim \mathcal{N}(0,1) $$ ...


1

Your comment makes it sound like you will: Model each category as binomial, and Construct confidence intervals for each category individually. In the case this is what you plan to do, let me first say this is not appropriate. This will create confidence hypercubes (?) which will be to liberal in their coverage (i.e. they will include parts of parameter ...


1

As Xi'an points out in comments, the substitution $\xi=\sigma^{-2}$ is called for. \begin{align} & \int_0^\infty \sigma^2\sigma^{-n-1}e^{-n(s/\sigma)^2} \, d\sigma \\[10pt] = {} & \int^0_\infty \xi^{-1} \xi^{(n+1)/2} e^{-ns^2\xi} \left( \frac{-\xi^{-3/2}\, d\xi} 2 \right) \\ & \qquad \text{where } \xi = \sigma^{-2} \\[10pt] = {} & \frac 1 2 \...


1

I will illustrate how this 'discretization' of a continuous-time Markov process can be done, using a process that is a little simpler than your queue. Continuous-time Markov process. Consider a continuous time Markov process roughly modeling decay of performance of a machine and rate of its repair. States are $S = \{1,2,3\},$ where 1 = poor, 2 = fair, 3 = ...


1

You could a mixed effects models for an ordinal outcome variable. The two most popular models for ordinal data are the proportional odds model and the continuation ratio model. For the latter you can find a detailed example on how to fit the model and extract the category-specific probabilities in the vignette Mixed Models for Ordinal Data of the ...


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