# Tag Info

Accepted

### Why are survival times assumed to be exponentially distributed?

Exponential distributions are often used to model survival times because they are the simplest distributions that can be used to characterize survival / reliability data. This is because they are ...
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### From uniform distribution to exponential distribution and vice-versa

It is not the case that exponentiating a uniform random variable gives an exponential, nor does taking the log of an exponential random variable yield a uniform. Let $U$ be uniform on $(0,1)$ and let ...
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### Why are survival times assumed to be exponentially distributed?

To add a bit of mathematical intuition behind how exponents pop up in survival distributions: The probability density of a survival variable is $f(t) = h(t)S(t)$, where $h(t)$ is the current hazard ...
• 2,240
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Let's begin by answering the question as it stands. Then we can respond to some of the points raised in comments. The question wants you to make the following assumptions: The future will behave ...
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### From uniform distribution to exponential distribution and vice-versa

You almost have it back to front. You asked: "If $X$ has a uniform distribution, does it mean that $e^X$ follows an exponential distribution?" "Similarly, if $Y$ follows an exponential distribution,...
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### Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$

The proof is given in the Mother of All Random Generation Books, Devroye's Non-uniform Random Variate Generation, on p.211 (and it is a very elegant one!): Theorem 2.3 (Sukhatme, 1937) If we define ...
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### Fitting exponential (regression) model by MLE?

This has been answered on the R help list by Adelchi Azzalini: the important point is that the dispersion parameter (which is what distinguishes an exponential distribution from the more general Gamma ...
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### Does the sum of two independent exponentially distributed random variables with different rate parameters follow a gamma distribution?

If $X\sim \exp(\lambda_1)$, $Y\sim\exp(\lambda_2)$ and $\lambda_1\neq\lambda_2$, the sum $Z=X+Y$ has pdf given by the convolution \begin{align} f_Z(z) &=\int_{-\infty}^\infty f_Y(z-x)f_X(x)dx \\&...
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### Relationship between poisson and exponential distribution

The other answers do a good job of explaining the math. I think it helps to consider a physical example. When I think about a Poisson process, I always come back to the idea of cars passing on a road. ...
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### Why are survival times assumed to be exponentially distributed?

You'll almost certainly want to look at reliability engineering and predictions for thorough analyses of survival times. Within that, there are a few distributions which get used often: The Weibull (...
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### Lambda - Exponential vs. Poisson Interpretation

Suppose I am waiting for a bus at a stop. And suppose that a bus usually arrives at the stop in every 10 mins. Now I define λ to be the rate of arrival of a bus per minute. So, λ = (1/10). Now I want ...
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### a fast uniform order statistic generator

The R code means returning $$(E_1,E_1+E_2,\ldots,E_1+\cdots+E_n)\Big/\sum_{i=1}^{n+1} E_i$$ and the result follows from checking that the differences between the cumulated sums of exponentials ...
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### Mean of inverse exponential distribution

Given that the inverse exponential distribution has $\alpha = 1$, you have stumbled upon the fact that the mean of the inverse exponential is $\infty$. And therefore, the variance of the inverse ...
• 13.5k
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### The Number of Exponential Summands in a Fixed Interval is Poisson

I propose taking, as a point of departure, the concept of a homogeneous Poisson process. This is a point process on the line (often thought of, and referred to, as a "time" line). The realizations ...
• 328k
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### Python - Test if my data follow a Poisson/Exponential distribution

One way to do what you're trying to do, is to compare your data with the hypothesized distribution (Exponential, Poisson, ..) and see if you can make any conclusions based on that comparison. Here is ...
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### What is the objective function to optimize in glm with gaussian and poisson family?

summary Linear model with least squares (Gaussian distributed observations) fit_lm = lm(log(World) ~ days, last_14) $$\sum_{\forall i} (\log(y_i) - X_i \beta)^2$$ ...
• 82.5k
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### Generating random samples obeying the exponential distribution with a given min and max

You describe truncation to an interval. I will elaborate. Suppose $X$ is any random variable (such as an exponential variable) and let $F_X$ be its distribution function, $$F_X(x) = \Pr(X\le x).$$ ...
• 328k
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### How are sums and differences of independent Exponential random variables distributed?

Because the characteristic function of an exponential distribution is $$\phi(t) = \frac{1}{1 - it},$$ when $X_1,\ldots, X_4$ are independent exponentially distributed variables (all with the same rate,...
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