1
vote
1answer
34 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
0
votes
0answers
63 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
1
vote
1answer
42 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
1
vote
2answers
34 views

Poisson approximation to binomial distribution: $f(x)\geq g(x)$ or $f(x) \leq g(x)$

We have a random variable $X$ which has a Binomial Distribution Bin(n,p) and a random variable $Y$ which has a Poisson Distribution Poisson(np). We are interested in $$f(x):=Pr[X \geq x].$$ For ...
1
vote
1answer
98 views

Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
0
votes
0answers
17 views

approximating a probability density

Let $f(x)$ be the probability density of a random variable $X$. Let the support of $f(x)$ be positive reals. If $f(x)$ is sufficiently smooth then one can approximate it with its Taylor series cut off ...
0
votes
0answers
46 views

Probability of a sample from a random variable with Gaussian distribution

I am studying a paper [1] which states that, as far as I understand, the probability of a single sample $x$ taken from a random variable $X$ with Gaussian distribution equals the Gaussian distribution ...
1
vote
1answer
73 views

Normal approximation for binomial distribution isn't giving correct result, z score comes out 0

I'm trying to use the normal distribution to calculate approximate values for the (cumulative) binomial distribution with large values (since it's impractical to evaluate the factorials). I'm very ...
1
vote
0answers
107 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
1
vote
0answers
277 views

How to calculate probability with sigmoid output in feedforward neural network?

first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem. I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
3
votes
0answers
100 views

Useful approximation of the pdf

Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
1
vote
0answers
114 views

Approximate CDF of the sum of a gaussian and a truncated gaussian

I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation ...
1
vote
0answers
209 views

Polynomial approx to the Normal density

I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF? Thanks $^{(1)}$ For example, some are ...
2
votes
1answer
679 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
1
vote
0answers
77 views

Combining multiple posterior distributions

I am new to Bayesian statistics, and thus have problems to come up with a solution for the following problem: Using Approximate Bayesian Computation (ABC), I generate a posterior distribution from ...
2
votes
1answer
222 views

Polynomial approximation of $\chi^2$ distribution pdf

The $\chi^2$ distribution PDF is $$f_{\chi^2}(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} \mathrm{e}^{-x/2} \mathbf{1}_{x \geq 0}$$ I am trying to find a polynomial approximation to this density ...
1
vote
1answer
162 views

bound of Erlang distribution

Is there any known polynomial bound of the Erlang distribution? I'd like to say that, given $k$ and $\lambda$ with probability p the r.v. is going to be less than some value x.
1
vote
0answers
428 views

Approximate linear density function for a normal distribution

I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where ...