# Tag Info

Accepted

### Intuitive explanation for the fat tails of the t-distribution

The heavy tails arise because in small samples, standard errors near zero have appreciable chances of arising and they are not associated with concomitantly small sample means that would cancel them ...
• 328k
Accepted

### Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

The fact that the distribution of $X-Y$ has mean $0$ does NOT imply that $\mathbb{P}(X-Y<0)$ and $\mathbb{P}(X-Y>0)$ are equal. So for the last step it is not enough that $X-Y$ has mean $0$, you ...
• 266

• 40.6k
Accepted

### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$,...

### What is the cross product of two probability distributions $P \times Q$?

I haven't read the specifics of the paper in question but the author meant by $P_x\times P_\epsilon$ the product measure. This measure is used when you have to equip a measure to a new measurable ...
• 9,337
Accepted

### Why can a model's SHAP values change on a new dataset?

Why can a model's SHAP values change on a new dataset? WHY SHOULDN'T THEY? When you calculate a mean on a new data set, you expect to get a slightly (or radically) different value. When you calculate ...
• 65k
Accepted

### Why is E(θ / (1 - θ)) different than E(θ) / (1 - E(θ))?

Taking an expectation does not commute with all arithmetic operations. While $E(X+Y)=EX+EY$, such a "distributivity" does not hold for other operations. For instance, the expectation of a ...
• 128k

### Is the realization of random variable also a random variable?

Constants can be considered as random variables if we like When dealing with probability theory and random variables, there are two things we might mean by a "constant". One thing we might ...
• 129k
Accepted

### What is the probability of selecting couples from a waiting room?

There are three kinds of couples after the selection: those from whom one individual was picked, those where both were picked, and those where neither were picked. When $k$ is the number of ...
• 328k

### Why does re-scaling my density plot using counts change the y-axis so much?

The difference between the two figures is that one displays the probability density and the other the count density. The total probability is always one (by definition). But the total counts can be ...
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### Estimating Probability Density for Sample

You are choosing distributions that match the empirical distribution. This will result in smooth estimates but the precision will be inherited from the variance of the empirical estimates. So there ...
• 95.7k

### (THEORY) Do Tree models output probabilities?

Your assessment of the situation is excellent. I would just add that in my practice random forests suffer some of the worst miscalibration that I’ve ever witnessed as a statistician. Even a single ...
• 95.7k

### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

Assuming you mean evaluate rather than solve*, the Tukey lambda distributions have easily evaluated quantile functions but the cdf doesn't have closed form https://en.wikipedia.org/wiki/...
Accepted

### On the estimated formula of covariance of two random variables

The two formulas aren't the same. The first formula is the formula for population covariance, denoted by $Cov(X,Y)$ (and also $\sigma_{XY}$): $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$ Suppose if the ...
• 660