0
votes
1answer
60 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
1
vote
1answer
37 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
42 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 ...
2
votes
1answer
85 views

Simplification trick

it is maybe at bit of a silly question, but one of our professors wrote the following equations and I would like to know what exactly he did. I'm sure it is something easy but I have no clue: ...
1
vote
1answer
104 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 ...
1
vote
3answers
33 views

An approximation: 2*number of events = twice the probability

My statistical mechanics textbook uses an approximation to derive a well-known result. The approximation is: Suppose for an event, the probability of an outcome is P. For n events, the probability of ...
0
votes
1answer
45 views

What is a good approximation of $(1-p)(1-q)$ as $(1-x)^2$, for $p,q \in (0,1)$?

I'm doing some scientific modeling, and I want to use $(1-x)^2$ to approximate $(1-p)(1-q)$, with $p, q \in (0,1)$. $p$ and $q$ are probabilities, and are not near zero. My intuition is that since ...
0
votes
0answers
50 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 ...
0
votes
1answer
25 views

How to find sequence from set of values

I have set of probability values arranged in ascending order, p1<p2<p3<...<pM. Now I want to assign set of numbers in the same manner in which probabilities are increasing. It means I ...
1
vote
0answers
37 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
0
votes
1answer
30 views

normal as approximation to binomial

Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of ...
0
votes
1answer
81 views

Approximation for expected number of distinct values

When I draw n evenly distributed integer random numbers from an range of [0,m], what is the expected number of distinct values? I am aware of this answer, but is look expensive to compute. Is there ...
1
vote
0answers
154 views

Discrete approximation to a continuous probability density function

I want to approximate a continuous, finite probability density function, with a specified number $N$ of points, in the following way: If the pdf is 1-dimensional, defined over the section [0,1], then ...
8
votes
3answers
452 views

the following inequality is true, but I can't prove it

The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find ...
1
vote
0answers
283 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 ...
0
votes
0answers
34 views

Product of Standard uniform & CLT

Suppose that $U_{i},\dots,U_{n}$ are iid $U(0,1)$. Use the central limit theorem to find an approximation for: $$P(U_{1}\times U_{2} \times\dots\times U_{25}\leq 6\times 10^{-6} )$$ Answer: Using ...
0
votes
1answer
193 views

Approximating probability of success of Bernoulli trials using KullbackÔÇôLeibler divergence

In "Probabilistic Graphical Models" book by Daphne Koller and Nir Friedman they have the following approximation of probability of r successful outcomes of N Bernoulli trials: $P(S_N=r)\approx ...
3
votes
1answer
128 views

What is the meaning of “mean-field”?

In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
3
votes
1answer
158 views

Brownian motion and hitting frequency

Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
3
votes
1answer
104 views

probability involving matching of discrete shapes on a square grid

Figure F exists on a regular square grid. T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees. A larger background grid of X by Y contains ...
1
vote
1answer
720 views

Approximating Coins Flips Problem

Approximate the probability of getting 500 heads out of a 1000 coin flip of unbiased coins to be within 5% of its true value (without the use of a calculator). I know that an exact probability ...
2
votes
1answer
692 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 ...
3
votes
1answer
104 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
1
vote
0answers
76 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...
0
votes
1answer
115 views

Approximation to $\mathbb{E}(X/Y)$

Let $X,Y$ are two random variables which are not necessarily independent. It is easy to get $\mathbb{E}(X)$ ann $\mathbb{E}(Y)$. I want to know: is there some approximation to ...
0
votes
0answers
70 views

How trustworthy is this kind of approximation?

I need to speed up an algorithm that takes as input the number of chips of several players in a tournament. Their chips can be seen as a normally distributed curve. Since my algorithm can handle a ...
3
votes
1answer
182 views

Calculate $E[X]$ using polynomial approximation of CDF

I have a black box called $F(t)$ ($~$($P~(X\le t)~$, $X$ is random variable) with me where I don't have any information on the exact expression of $F(t)$. But if I supply a $t\ge 0$ I will get a value ...
3
votes
2answers
80 views

Approximation or bound of $\operatorname{Pr}(X<\operatorname{E}(X))$

$X$ is a continuous random variable (we can assume some statistic (e.g., mean and variance) are known, but the distribution is unknown). Consider a probability ...
1
vote
2answers
962 views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
2
votes
2answers
152 views

Generate a Monte Carlo sample from a PDF defined by a Fourier Series

I have a probability distribution (PDF) defined by a Fourier series.. actually it's a purely cosine series over a known range. The PDF quite smooth, so most of the power is in the low 5 or so ...
3
votes
1answer
697 views

Stirling's Approximation and Binomial Random Variable

I am trying to follow equation (1.13) in MacKay's Information Theory textbook (http://www.inference.phy.cam.ac.uk/itprnn/book.pdf). It is: $$ \ln \binom{N}{r} = \ln \frac{N!}{(N-r)! r!} \approx (N-r) ...
19
votes
2answers
550 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
2
votes
0answers
204 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
1
vote
0answers
432 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 ...
3
votes
3answers
690 views

How to evaluate probability estimators with only external information?

Here's a problem that I have pondered over many times without ever coming to a satisfactory solution: Let's say that we have a series of random events: V(i) for I = 1 to n. Each of these events will ...
4
votes
1answer
459 views

Large Deviation Properties of a function of a geometric random variable

Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is $p_s = (1 - x)^{s - 1} x$, Consider the following function of ...
10
votes
3answers
414 views

Solving randomized recurrence relation

I'm looking at the random sequence $x_n$ with $x_0=x_1=1$ and \begin{equation} x_{n+1}=2x_n\pm x_{n-1} \end{equation} where we choose the $\pm$ sign independently with equal probability. Now ...