# Tag Info

Accepted

### What exactly is the alpha in the Dirichlet distribution?

The Dirichlet distribution is a multivariate probability distribution that describes $k\ge2$ variables $X_1,\dots,X_k$, such that each $x_i \in (0,1)$ and $\sum_{i=1}^N x_i = 1$, that is parametrized ...
• 140k
Accepted

• 25.1k

### How to decide which glm family to use?

Generalized linear model is defined in terms of linear predictor $$\eta = \boldsymbol{X} \beta$$ that is passed through the link function $g$: $$g(E(Y\,|\,\boldsymbol{X})) = \eta$$ It models ...
• 140k
Accepted

### Why are log probabilities useful?

The log of $1$ is just $0$ and the limit as $x$ approaches $0$ (from the positive side) of $\log x$ is $-\infty$. So the range of values for log probabilities is $(-\infty, 0]$. The real advantage is ...
• 52.4k
Accepted

### Does mean=mode imply a symmetric distribution?

Mean = mode doesn't imply symmetry. Even if mean = median = mode you still don't necessarily have symmetry. And in anticipation of the potential followup -- even if mean=median=mode and the third ...
• 286k
Accepted

### "Absolutely continuous random variable" vs. "Continuous random variable"?

The descriptions differ: only the first one $(*)$ is correct. This answer explains how and why. Continuous distributions A "continuous" distribution $F$ is continuous in the usual sense of ...
• 328k
Accepted

### I know the 95% confidence interval for ln(x), do I also know the 95% confidence interval of x?

That is a 95% confidence interval for $x$, but not the 95% confidence interval. For any continuous strictly-monotonic transformation, your method is a legitimate way to get a confidence interval for ...
• 129k
Accepted

### Expectation of 500 coin flips after 500 realizations

If you "know" that the coin is fair then we still expect the long run proportion of heads to tend to $0.5$. This is not to say that we should expect more (than 50%) of the next flips to be ...
• 8,457
Accepted

### Intuitive explanation of Kolmogorov Smirnov Test

The Kolmogorov-Smirnov test assesses the hypothesis that a random sample (of numerical data) came from a continuous distribution that was completely specified without referring to the data. Here is ...
• 328k
Accepted

### Which distribution has its maximum uniformly distributed?

Let $F$ be the CDF of $X_i$. We know that the CDF of $Y$ is $$G(y) = P(Y\leq y)= P(\textrm{all } X_i\leq y)= \prod_i P(X_i\leq y) = F(y)^n$$ Now, it's no loss of generality to take $a=0$, $b=1$, since ...
• 41.9k
Accepted

### Derivation of change of variables of a probability density function?

Suppose $X$ is a continuous random variable with pdf $f$. Let $Y=g(X)$, where $g$ is a monotonic function. The function $g$ could be either monotonically increasing or monotonically decreasing. If $g$ ...
• 2,107

### Could any equation have predicted the results of this simulation?

At any given point in the game, you're $3$ or fewer "perfect flips" away from winning. For example, suppose you've flipped the following sequence so far: $$HTTHHHTTTTTTH$$ You haven't won ...
• 571

### What are the properties of a half Cauchy distribution?

A half-Cauchy is one of the symmetric halves of the Cauchy distribution (if unspecified, it is the right half that's intended): Since the area of the right half of a Cauchy is $\frac12$ the density ...
• 286k
Accepted

### Is Cauchy distribution somehow an "unpredictable" distribution?

While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may ...
• 286k

### Is a distribution that is normal, but highly skewed, considered Gaussian?

A distribution that is normal is not highly skewed. That is a contradiction. Normally distributed variables have skew = 0.
• 125k
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 ...
• 1,401
Accepted

### How to generate random integers between 1 and 4 that have a specific mean?

I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering. Because the product of the sample mean ...
• 328k
Accepted

### Can statistical units measured per thousand inhabitants be bigger than 1000?

This is not a rate per one thousand people, this is the absolute number of people, with one unit equating 1,000 people. So if you see something like 3,258.1, it simply means 3,258,100 people. This is ...
• 4,924
Accepted

### I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

I would like to offer a very simple, intuitive explanation. It amounts to looking at a picture: the rest of this post explains the picture and draws conclusions from it. Here is what it comes down to: ...
• 328k

### Are probabilities preserved under function transformation?

No. Take $X$ uniform on $[-1,1]$ and $a=0$. Then $\Pr(X < a ) = 1/2$. On the other hand $\Pr(X^2<a^2) = 0$.
• 19.5k

### Statistical interpretation of Maximum Entropy Distribution

This isn't really my field, so some musings: I will start with the concept of surprise. What does it mean to be surprised? Usually, it means that something happened that was not expected to ...
• 81.3k
Accepted

### 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 ...
• 286k

### What is the advantages of Wasserstein metric compared to Kullback-Leibler divergence?

Wasserstein metric most commonly appears in optimal transport problems where the goal is to move things from a given configuration to a desired configuration in the minimum cost or minimum distance. ...
• 4,299
Accepted

### Are differences between uniformly distributed numbers uniformly distributed?

No it is not uniform You can count the $36$ equally likely possibilities for the absolute differences ...
• 40.6k
Accepted

### Distribution that has a range from 0 to 1 and with peak between them?

One possible choice is the beta distribution, but re-parametrized in terms of mean $\mu$ and precision $\phi$, that is, "for fixed $\mu$, the larger the value of $\phi$, the smaller the variance of $y$...
• 140k
Accepted

### The sum of independent lognormal random variables appears lognormal?

This approximate lognormality of sums of lognormals is a well-known rule of thumb; it's mentioned in numerous papers -- and in a number of posts on site. A lognormal approximation for a sum of ...
• 286k
Accepted

### Definition of sample space

The basic intuition is that: $\Omega$ is the set of outcomes that can happen. $\mathcal S$, a $\sigma$-field of subsets of $\Omega$, represents what information is available. It represents what ...
• 22.7k
This holds only if $f$ is monotonically increasing. If $f$ is monotonically decreasing, then $P(f(X)\leq f(a)) = P(X \geq a)$. For instance, if $f(x) = -x$, and X is a normal die roll, then \$P(X \leq ...