# Tagged Questions

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### Proof of “Normal approximation to the log-normal distribution” [closed]

I saw the post about the normal approximation to lognormal (Normal approximation to the log-normal distribution). The proof is shown as well. Yet as I'm looking for the proof in a journal article form ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
208 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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.
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 ...