2
votes
1answer
29 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
0
votes
0answers
19 views

How to estimate $\sum_{x=1:n}{xf(x)}$ having $\tilde{f}$

I have an estimator $\tilde{f}(x)$ whose error is at most $\epsilon$, i.e., $\frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \leq \epsilon$. I want to estimate $\sum_{i=1:n}i.f(i)$ with a small error. But if I ...
4
votes
1answer
63 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
14 views

regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error term. We know covariance matrix ∑ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would ...
0
votes
0answers
13 views

Estimating the accuracy of precipitation odds.

Modern weather forecasts often include a percentage chance of rain. If this estimate was a constant it would be a simple matter of calculating the ratio. But the reality is quite another. This is the ...
0
votes
0answers
15 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers
1
vote
1answer
69 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
0
votes
0answers
50 views

Probabilities and Estimation of average and standard deviation

I've done a good bit of this number but I have trouble with part 2. I'll show you my work and the questions I can't figure in bold. A guy has a machine that scans his apples. The machine rules are : ...
0
votes
1answer
23 views

How to call $(E\hat{x} - x)^2$?

Let $\hat{x}$ be an estimation of $x$. Quantity $E(\hat{x} - x)^2$ is called Mean Squared Error. How one would call $(E\hat{x} -x )^2$?
0
votes
0answers
71 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
2
votes
4answers
202 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
1
vote
0answers
49 views

Calculating a metric to compare multiple posterior probability distributions

I am beginner in mathematics/statistics and apologise in advance for my faulty use of language. Especially because I assume this to be a simple problem. I am working on a problem in statistical ...
0
votes
0answers
427 views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
0
votes
0answers
37 views

What is $E[T^2]$ of $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$ ??

What is second moment i-e $E[T^2]$ of random-variable: $T=\frac{1}{N}\sum\limits_{n=1}^N (X[n]+W[n])^2$, Where $X[n]$ and $W[n]$ are both 'independent' of each other and 'stationary'. Moreover, ...
0
votes
1answer
34 views

How to find $X_i$ from this equation

Suppose $X_i=\nu_i+\frac{m-i}{m}X_{i+1}+\frac{i}{m}X_{i-1},\quad 1\le i\le m$ where $X_0=X_{m+1}=0$. I need to find an expression for $X_i$ in terms of $v_i$, $i$, and $m$. I know how to find it ...
0
votes
2answers
29 views

Calculating a sample's representativeness to confirm/refute a given hypothesis?

Why hello! I'm fairly new to statistics, which is why I'm somewhat confused as to how I can approach this problem in a scientific way. The problem: Experiments are conducted to find the probabilities ...
3
votes
1answer
48 views

Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
0
votes
0answers
21 views

Density estimation for two sets of samples

Imagine two sample sets $A$ of size $\#A$ and $B$ of size $\#B$. First density estimation for both is done separately which yields $P_A$ and $P_B$ based on those densities. In what way is ...
0
votes
0answers
40 views

Maximization of The Likelihood Function of Vector Entries and Its Norm

I'd be happy for assistance with the maximization of the likelihood function of the following model. The Parameters Vector $ \mathbf{\Theta} = [{x}_{1}, {x}_{2}] $. The measurement vector is $ ...
1
vote
0answers
19 views

How do I compute the variance (or confidence interval) of a Maximum Spacing estimator?

I am trying to solve a problem using a Maximum Possible Spacing estimator (see Maximum spacing estimation on wikipedia for links). Details on what I am trying to do can be found in the following ...
1
vote
1answer
65 views

Advanced urn problem

Imagine there are two urns — urn A and urn B. Urn A contains 3 blue balls and 7 red balls. Urn B contains 7 blue balls and 3 red balls. Balls are now randomly drawn from one of these urns where the ...
2
votes
1answer
136 views

Likelihood of a Uniform Distribution

I have been looking at this solution for two days and still can't understand the solution. The question is as follows: Given $w[i], i = 1, 2, \ldots, N$ are IID following a distribution of $U[0, ...
0
votes
1answer
126 views

square root estimator

Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots. This form also seems ...
0
votes
0answers
49 views

How to find conditional distribution of outputs given input

Suppose we have a training dataset $x_1, y_1, \dots, x_T, y_T$. Our goal is to empirically estimate $P(y|x)$ using histogram of data. What histogram should I exactly use? One option is to use the ...
0
votes
1answer
120 views

What is $\sum\ln{(x_i!)}$?

I started learning statistics and in my homework i should find the Maximum Likelihood Estimate. The function is $f_x(x)=e^{-\lambda n}\prod_{i=1}^n \frac{\lambda^{x_i}}{x_i!}$ Now i take the ...
2
votes
1answer
289 views

Method of Moments and Maximum Likelihood estimators?

The random variables $X_1,...X_n$ are independent draws from continuous unifirm distribution with support $[0,\theta]$. Derive a method of moments and maximum likelihood estimators of $\theta$. Your ...
2
votes
1answer
98 views

What is the probability of the number 1 and number 2 employees getting the bonus at a call center?

Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it. Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and ...
1
vote
1answer
174 views

$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$

$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$ $\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$ $\Rightarrow \log ...
2
votes
2answers
45 views

Estimating Poisson $\theta$ only from which percentage of intervals have events

Radioactive particles are emitted randomly over time from a source at an average rate of per second. In $n$ time periods of varying lengths $t_1,t_2,\dots,t_n$ (seconds), the numbers of particles ...
0
votes
1answer
980 views

Finding an unbiased estimator for the negative binomial distribution

Consider a negative binomial random variable Y as the number of failures that occur before the r th success in a sequence of independent and identical success/failure trials. The pmf of $Y$ is ...
1
vote
0answers
60 views

Fast way to estimate cardinal number of subset

I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets, $ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
1
vote
2answers
41 views

Expected value of total accumulated lifetime (understanding gap in proof)

Problem: I understand the first line $E(T) = ...$ However, I don't get the next two steps. I feel like I almost get it. It's like we are factoring out a $\sum_{j=1}^{20}$ but how did he ...
0
votes
1answer
58 views

Likelihood of the mean of one random variable with unknown parameters greater than another

Assume we have two random variables $X$ and $Y$ that are gamma distributed (or normally distributed, if it makes the math easier) with unknown parameters. We have samples $x_1,x_2,...,x_m$ and ...
0
votes
1answer
37 views

Approximation question

Cars and buses arrive at a bridge according to the independent Poisson processes at a rate of $3$ cars/minute and $1$ bus/ minute. What is the chance that strictly more buses arrive than cars in a ...
0
votes
2answers
49 views

Correlation bound

Let x and y be two random variables such that: Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An ...
0
votes
1answer
25 views

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define?

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define ? (Is CrossValidated better place for this question ?)
1
vote
2answers
82 views

Independent tests bound. (Chernoff/Azuma?)

I have a series of N Bernoulli tests (p, 1-p). I need to calculate a probability of passing more than N/2 tests, depending on N and p. The obvious solution is Chernoff bound: $\varepsilon \leq ...
0
votes
1answer
50 views

System of equations: Can I solve this system of equations?

I want to ask you which field of mathematics contains following kind of problem. Suppose that following equations are given $\alpha\times x_{1}=C_{1}$ $\alpha\times x_{2}=C_{2}$ $\alpha\times ...
2
votes
0answers
92 views

Approximability of continuous sampling by discrete sampling - definitions?

Are there standard definitions that express the notion that sampling from a given continuous random variable can be approximated "to any desired degree of accuracy" by sampling from an appropriately ...
0
votes
1answer
146 views

Continuously sampled event: Estimating the value of a future data point, based on past measurements and their tendency

Problem I'd appreciate some ideas on how to define a formula to estimate the value of a future data point for a continuously sampled event, based on past measurements and their tendency. At any ...
1
vote
2answers
71 views

Estimation with non-independent errors

I have the following model: $Y_1=\beta+\varepsilon_1+\varepsilon_2$ $Y_2=\beta+\varepsilon_3+\varepsilon_4$ $Y_3=\beta+\varepsilon_1+\varepsilon_4+\varepsilon_5$ ...
0
votes
1answer
281 views

Expected value for the number of goals in a game

I'm trying to use odds data from bookmakers to estimate the expected number of goals in a game. We have these known facts: P(o4.5) = 0.573 P(o5.5) = 0.458 P(o6.5) = 0.279 P(o4.5) is the ...