9

Both the normal and uniform distributions are continuous; ie, any particular value has probability of zero. Obviously there is numerical precision and other considerations involved with a machine-specific implementation but for all intents and purposes, you can suppose that $\mathbb P(X = x) = 0$ for any particular $x$; ie, the probability that you randomly ...


6

For "with replacement" and "without replacement" to be distinct, you'd need a finite population. If you have a finite population, you don't have a normal distribution (nor any other continuous distribution). [Implementation wise, however, there's only a finite number of different values that it's possible to generate on a computer -- at any fixed number of ...


6

This is an interesting observation that I am able to reproduce in lme4 (version 1.1-21). I have also implemented the Gamma mixed model in my GLMMadaptive package (version 0.6-9 available currently on GitHub), which seems to recover the correct value for the standard deviation of the random intercepts. The following code illustrates the issue and compares the ...


4

Maybe this is just an approximation, and this distribution converges on p(v) for large samples? No, it's exact for any size of sample. The joint distribution is the product of the known marginal and the conditional: $f_{U,V}(u,v) = f_{V|U}(v)\cdot f_{U}(u)$ So we can generate a sample of observations from the joint distribution of $(U,V)$, that is, a set ...


4

Simply invert the distribution function. It's efficient. Here are the details. A standardized Lomax distribution with shape parameter $\alpha\gt 0$ has distribution function (CDF) $$F(z;\alpha) = 1 - \frac{1}{(1+z)^{\alpha}},\quad x \ge 0.$$ Its quantile function is thereby straightforward to find as $$F^{-1}(q;\alpha) = \left(1 - q\right)^{-1/\alpha}-...


4

Once you have the form of the PDF, there are various techniques for sampling. Some easy forms can be handled via Inverse Transform Sampling. Some special forms can be handled via methods special methods, e.g. sampling from normal distribution via Box-Müller. Other general methods exist for PDFs with non-easy/non-special forms (i.e. inverse transform sampling ...


4

For what you described, I cannot see any direct relationship with MCMC. What you needed is just a forward sampling. Here is how it works (suppose we have discrete binary random variables): Step 1. get a sample for $X_1$. In order to do this step, we need to have the distribution $P(X_1)$. (Something like $$ P(X_1)=\left\{ \begin{array}...


3

All coins are biased. It's a question of how biased they are, but no coin has exactly equal chances of being heads or tails. In the frequentist approach, I think this calls for a test of equivalence: Decide how close to 0.50000 you will accept as unbiased. Do a power analysis to determine how many flips you will need to have a good (0.80? 0.90?) chance of ...


2

The problem seems to be with how you are interpreting the family-wise error rate. There are three tests being conducted simultaneously. Let $E_i$ be the event that hypothesis $i$ is rejected, $i=1,2,3$. In your simulations, all null hypotheses are true, so that $$P(E_i) = 0.05, \ i=1,2,3.$$ What your code is doing. You are running $M=1000$ simulations, ...


1

If you're using your current predictions to influence the estimation of your future predictions, you get a positive feedback loop and wave goodbye to any real-world distribution. For example, an initial transition matrix $[[0.99, 0.01], [0.01, 0.99]]$ will asymptotically wind up predicting either only As or only Bs, depending on your start state.


1

You don't need to simulate. Let $n$ be the no. observations in each group; by definition $n$ observations overall exceed the median of the pooled observations, & $n$ fall short of it. The data therefore constitute a contingency table with fixed row margins $(n, n)$ & fixed column margins $(n, n)$; the count in any one cell determines the counts in ...


1

If I understand correctly, you want to generate new samples based on a PDF determined from your data. Such a process is known as oversampling and there exists a paper which does exactly this. Check it out: https://ieeexplore.ieee.org/abstract/document/6252384


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