# Questions tagged [probability-distributions]

For questions related to probability distributions, functions that relate a given value to the likelihood that a random variable will take that value.

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### Markov Chain - Stationary or Borderline distribution [closed]

I have a Markov chain problem below, where : The problem An urn initially contains 3 black balls and 1 red ball. The balls are indistinguishable to the touch. One ball is randomly drawn. If this ball ...
69 views

### How to calculate Cycle & Safety Stock

I have been asked to calculate cycle and safety stock levels for one of our business units. I have very little/no knowledge on the subject of inventory theory so would like to ask what a reasonable ...
120 views

### Convexity of the variance of a mixture distribution

$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words: $$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$ where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
114 views

### What is the meaning of monotone hazard rate (MHR) distribution?

It might be somewhat irrelevant to this forum but I think that many people here are familiar with this concept. I have seen that many papers assume that customers' valuation ($F$) is a monotone hazard ...
215 views

### Safety stock for log-normal distribution demand

I came across this example on how to model your lead time demand as a log-normal distribution and calculate the safety stock. https://www.linkedin.com/pulse/why-you-keep-missing-your-service-level-...
Suppose $X_1,\ldots,X_n$ are drawn i.i.d from a uniform distribution on $[0,1]$ and let $x$ be the random vector $(X_1,\ldots,X_n)$. Then consider the random variable $Y_v = v^\top x$ for all $v \in \... 1answer 84 views ### Question about a queueing problem Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of phone calls is assumed to be distributed exponentially, ... 1answer 218 views ### Standard cumulative distribution function with optimization model variable We all know that expressions in mathematical optimization models can't contain "black boxes" around a decision variable since everything has to be written using mathematical expressions. For example, "... 1answer 272 views ### Convexity of Variance Minimization$X$is a discrete random variable taking value$x_n$with probability$1/N$for$n=1, \ldots,N$. I would like to set the$x_n$values in an optimization problem. My objective is to minimize the ... 1answer 95 views ### Model or State Uncertainty in Queueing Model due to uncertain arrival rate Introduction I am currently modelling a scenario where two queues need to be served by a single server in a non preemptive discipline. I am quite sorted on generating the optimal policy via Value or ... 4answers 855 views ### Modeling the uncertainty of the input parameters There are many approaches to deal with the uncertainty such as stochastic programming, robust optimization and fuzzy programming. Finding a suitable approach that is applicable in the real situations ... 2answers 220 views ### Queuing models in R,$\lambda$Little It's noted that the number of folks in a stationary system will maintain an average equal to the rate of arrival multiplied by the mean of the service distribution. The formula$L = \lambda w$is ... 1answer 208 views ### How to fit a Beta distribution to three estimates from an “expert”? I'm modeling a process time,$X$, for a simulation study and have an "expert" estimate of the minimum,$\hat a$, the most likely (mode),$\hat m$, and the maximum,$\hat b$. I'd prefer to avoid the ... 1answer 597 views ### Loss functions for specific probability distributions? For a random variable$X$with pdf$f(x)$, the loss function* is defined as $$n(x) = \mathbb{E}[(X-x)^+] = \int_{x}^\infty (y-x)f(y)dy,$$ where$a^+ = \max\{a,0\}$. Or, for a discrete distribution,$...
In a continuous-review $(r,Q)$ inventory system under a type-1 service level constraint, if the demand per unit time is distributed as $N(\mu,\sigma^2)$ and the lead time, $L$, is a constant, then the ...