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Results tagged with distributions
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user 2970
A distribution is a mathematical description of probabilities or frequencies.
28
votes
How to identify a bimodal distribution?
There is a well-known paper by Silverman that deals with this issue. It employs kernel-density estimation. See
B. W. Silverman, Using kernel density estimates to investigate multimodality, J. Roya …
- 27.4k
16
votes
Accepted
Is it possible to analytically integrate $x$ multiplied by the lognormal probability density...
Short answer: No, it is not possible, at least in terms of elementary functions. However, very good (and reasonably fast!) numerical algorithms exist to calculate such a quantity and they should be pr …
- 27.4k
19
votes
Accepted
Closed form expression for the quantiles of $\alpha_1\sin(x)+\alpha_2\cos(x)$
Quantiles of a trapezoidal distribution
Let $Y = X_1 + X_2$ where $X_1$ and $X_2$ are independent $\mathcal U(-a,a)$ and $\mathcal U(-b,b)$ distributions. …
- 27.4k
11
votes
Accepted
Anscombe transform and normal approximation
Here is a sketch of a proof which combines three ideas: (a) the delta method, (b) variance-stabilization transformations and (c) the closure of the Poisson distribution under independent sums.
First, …
- 27.4k
15
votes
Accepted
Is the Poisson distribution stable and are there inversion formulas for the MGF?
The associated theory is more commonly developed for characteristic functions since these are fully general: They exist for all distributions without support or moment restrictions. …
- 27.4k
15
votes
How to estimate parameters for Zipf truncated distribution from a data sample?
MEJ Newman, Power laws, Pareto distributions and Zipf's law, Contemporary Physics 46, 2005, pp. 323-351.
M. … Mitzenmacher, A Brief History of Generative Models for Power Law and Lognormal Distributions, Internet Math., vol. 1, no. 2, 2003, pp. 226-251.
K. …
- 27.4k
10
votes
Accepted
Distribution of a ratio of uniforms: What is wrong?
Here is a hint.
Consider carefully the term $\mathbb P( X \leq z Y \mid z Y < 1 )$. In particular, for concreteness, choose $z = 2$, so that we are considering the event $\mathbb P( X \leq 2 Y \mid …
- 27.4k
10
votes
Accepted
What's the name of this discrete distribution (recursive difference equation) I derived?
In a sense, what you have done is characterize all nonnegative
integer-valued distributions. … specifically,
Proposition: A sequence $(p_n)$ taking values in $[0,1]$ determines a distribution on the nonnegative integers if and only
if $$
-\sum_{n=0}^\infty \log(1-p_n) = \infty \>, $$ and all such distributions …
- 27.4k
68
votes
Accepted
Fake uniform random numbers: More evenly distributed than true uniform data
Yes, there are many ways to produce a sequence of numbers that are more evenly distributed than random uniforms. In fact, there is a whole field dedicated to this question; it is the backbone of quasi …
- 27.4k
10
votes
Accepted
Mean and variance of log-binomial distribution
We can use an entirely analogous technique to the one typically used to calculate the moments of a lognormal.
In particular, note that if $\newcommand{\E}{\mathbb E}X \sim \mathrm{Bin}(n,p)$ and $Y = …
- 27.4k