0
votes
2answers
33 views

probability question needs some help

suppose $X$ and $Y$ are independent and identically distributed random variables that are uniformly distributed on $[0,1]$ What is the PDF of $ W=Y-X $ i tried to draw a picture to illustrated it ...
1
vote
1answer
12 views

just a manipulation problem for statistics

I am a little bit confused when i was asked to Suppose $X_1, X_2,\ldots,X_n$ is a simple random sample from a continuous distribution with density function $f(x)$. Consider the new random variable ...
0
votes
0answers
23 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
1
vote
1answer
24 views

What is the pivotal quantity

I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated. a)$ X_1,....X_n$ are uniform iid on the interval ...
0
votes
0answers
13 views

Calculate probability of total population from sample

I have a population of infinity size. Each Unit is Boolean(True/False). I take a random sample of size N of which M elements are True(and N-M are False). I'd like to know what the probability is that ...
0
votes
0answers
22 views

How to find MLE function?

I have two vectors of known values x and y. And the relationship between them is y=sin($\theta$$\cdot$x)+$\epsilon$, $\epsilon$~N(0,1) . The question is how i find the MLE function for $\theta$?
0
votes
0answers
5 views

Stationary point of Unnormalized and Normalized KL-divergence minimization

I have encountered a problem, basically related to the http://arxiv.org/abs/1206.6679 . If I want to minimize the normalized KL-divergence KL(Q||P) with Q a multivariate Gaussian distribution. but in ...
0
votes
1answer
40 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
0
votes
2answers
68 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
1
vote
1answer
35 views

asymptotic normality and central limit theorem

Here's the question Can somebody explain the difference between asymptotic normality and central limit theorem? They seem very similar to me.
0
votes
1answer
40 views

Fitting a function with relative uncertainties

I want to find the parameters of a function that better fit some measurements. Usually, I use least squares. With it I am assuming that the best function is the most likely one and that the residuals ...
1
vote
0answers
18 views

Confidence bounds given random verification

[Edits made for clarification and brevity.] I'm working on an idea for a fault detection algorithm, and I've boiled it down (I think) to the following problem. A box contains 10 balls. The balls can ...
0
votes
2answers
35 views

How to find confidence interval of 0.95 in this problem?

For sample $x_1,\cdots,x_{100}$, following holds. $\sum_{k=1}^{100}x_k=400$ and $\sum_{k=1}^{100}x^2_k=2500$. Find the confidence interval of 0.95 for the population mean $m$. I've calculated the ...
0
votes
0answers
25 views

Computation of conditional probabilities

I am aware this question might not be well formulated, but it is not very clear for me neither, so if anyone could help me explicit it... I observe many examples e_i. For each example, I compute two ...
0
votes
1answer
74 views

Queuing Theory with Poisson Distribution

Suppose customers arrive in a one-server queue according to a Poisson distribution with rate lambda=1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with ...
0
votes
2answers
30 views

investigating a relationship at $5 \% $ level significance

$n=12$ $\bar{x}=? $ sample average = $\sigma $ standard deviation = $\alpha =$ $H_a :$ Degrees of freedom if applicable Critical value(s) = Sample mean Standard error of mean = $\frac{\sigma } ...
0
votes
1answer
23 views

Finding P value

I have these observations $(2,3.2,3.8,2.5,3.3,2.8,3.0,3.4)$ from $X \sim N(\mu,\sigma^2)$ and i want to calculate the $P$-value testing $H_0: \mu =3.2$ against $H_1 \neq 3.2$ with $\sigma = 0.6$ ...
1
vote
1answer
105 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
0
votes
0answers
57 views

Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
0
votes
0answers
7 views

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$?

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$? I am in the statistical model $X_{hi} \sim ...
0
votes
1answer
18 views

Confidence intervals for mutliparameter fit

I am doing fits to some data and currently using mathematicas nonlinearmodel fit to generate fits and CIs, as well as doing bootstrapping for CIs for completeness. However, I would like to know how to ...
1
vote
1answer
32 views

Variance with correlated variables

A simple question that I don't manage to solve: I can use different methods to measure a magnitude $x$. The results of these methods are correlated and have some uncertainties. Combining the results ...
2
votes
3answers
92 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
0
votes
0answers
18 views

Efficient algorithm for point estimation of a dependent random variable

Suppose $X$ is a normal-distributed random variable and $f$ is a known smooth function (possibly quite complicated, with many oscillations). Let $p(y)$ be the pdf of the dependent random variable $Y = ...
2
votes
2answers
38 views

Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of ...
1
vote
0answers
25 views

Adjusting regression for small sample bias

I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc. I would like to test the null hypothesis of no ...
0
votes
2answers
51 views

$QQ$-plot - Why do we choose the empirical distribution $F_n(x) = \frac {\#\{y \in S \mid y \le x\}} n$, $S$ is sample, for comparison with normal?

$QQ$-plot - Why do we choose the empirical distribution $F_n(x) = \frac {\#\{y \in S \mid y \le x\}} n$, $S$ is sample, for comparison with normal ? Let $S$ be our sample of size $n$. Then we form ...
0
votes
2answers
27 views

integral support for a density function

$$f_{XYZ}(xyz)=ke^{-(x+y+z)}$$ $$ 0<x<y<z $$ I must find for which k this is a density function. $$\int_{Rx}\int_{Ry}\int_{Rz} ke^{-(x+y+z)} dzdydx =1$$ $$k\int_{Rx}\int_{Ry}\int_{Rz} ...
1
vote
1answer
78 views

Exponential family of distributions?

Consider the parametric class formed by the density functions defined as follows: $$ f(y,\theta) = \frac {2} {\Gamma (1/4)} e^{-(y-\theta)^4},\quad y\in\mathbb R,\quad\theta\in\mathbb R. $$ Does ...
1
vote
1answer
101 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
0
votes
0answers
17 views

Textbooks with worked examples on identifiability and regularity

I'm looking for textbooks that would have worked examples on identifiability and regularity, something similar to a Schaum's outline. Are there any textbooks that have worked examples on these ...
0
votes
1answer
28 views

Explain how $ p(a,b,c,d) = \frac{\phi(a,b,c)\phi(a,b,c)}{Z} $ leads to $ Zp(a,b,c) = \phi(a,b,c) \sum_d \phi(b,c,d) $

this might be a quiet basic question: Let $ \phi(\chi^i) $ be a potential. Then we have $ p(a,b,c,d) = \frac{\phi(a,b,c)\phi(b,c,d)}{Z} $ By summing we have: $ Zp(a,b,c) = \phi(a,b,c) \sum_d ...
1
vote
1answer
46 views

How can $n$ variables have $2n$ degrees of freedom?

Formally, if $Y_i\sim \mathrm{Exp}(\lambda)$, then $2\sum_{i=1}^n Y_i \sim \Gamma(n,2)$, which is the chi-squared distribution with $2n$ degrees of freedom. Intuitively, however, I think of degrees ...
0
votes
0answers
35 views

Neyman–Pearson lemma for non monotonic spaces

Question: Does the Neyman–Pearson lemma give instructions for how to construct the test when the outcome space is not monotonic? I suspect the answer is NO, but I would like to: Get an affirmative ...
0
votes
0answers
12 views

Logit Nomal Prior Distribution

$$\mu \sim N(\mu_0,\sigma_0)$$ $$ X_i \sim LN(\mu,\sigma_x)$$ Does anyone know any method for finding the posterior distribution $P(\mu|X)$ or at least any idea of how to estimate it numerically. I ...
0
votes
0answers
24 views

A name for this equality? $-E\left[ \frac{\partial U(h)}{\partial \phi}\right]=E[U(h)U(h_{opt})]$ $\forall h$

I have an estimating equation of the form $U(\phi;h)=m(X,Y)h(X)$. The $m(X,Y)$ part is fixed, but the $h(X)$ part can vary. I am trying to find out the optimal $h(X)$ so that the estimator has least ...
2
votes
1answer
67 views

Rolling standard deviations

I am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: ...
1
vote
1answer
66 views

Book for probability and various probability distribution functions.

Please suggest a book/books where i can understand Probability theory (with lots of example and solution) examples on permutations and combinations. list of all probability distribution functions, ...
3
votes
1answer
105 views

How do I 'reverse engineer' the standard deviation?

My problem is fairly concrete and direct. My company loves to do major business decisions based on many reports available on the media. These reports relates how our products are fairing in ...
0
votes
1answer
39 views

Fisher-Tippet Theorem: how to compute the limit

Here is the question: The PDF of $X_{1},X_{2}...X_{n}$ are $f_X(x)=1-e^{-x},x>0$. They are independent. Let $M_{n}=max(X_{1},X_{2}...X_{n})$ and ...
1
vote
0answers
95 views

Central Limit Theorem Clarification

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
0
votes
1answer
17 views

Confusion related to the tractability of an integral

I have this confusion related to the tractability of an integral. In the attachment given below for equation 3 why is it intractable. Further in equation 4 they have said that there are $K^n$ ...
1
vote
1answer
136 views

a distribution of a sqrt of a Normal distribution

i have a Normal(0,1)=X. and (X_{1},....X_{20}). I have to calculate the distribution of $T=\sqrt{|Z|}$ with Z= $\dfrac{1}{20} \sum_{1}^{20}X_{i}$ and his average. I have done this, but Im not very ...
1
vote
0answers
51 views

Properties of almost sure convergence

If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$, $\frac{1}{p}|S|_{F}^{2} ...
0
votes
0answers
11 views

Approximations for the Coarse Graining of the one norm difference of two probability distributions

I want to coarse grain $D(P_{1},P_{2}) = \frac{1}{2} \sum_{r}^{D} |Pr(r|1) - Pr(r|2) |$ for two distinct distributions Pr(r|0) and Pr(r|1). Such that $\sum_{r} P(r|1) = 1$ and $\sum_{r} P(r|2) = 1$. ...
3
votes
0answers
58 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
0
votes
0answers
35 views

Uniform choice for Prior Distribution

My prior function is $\Phi\left(\mathbf{k}_\ell,W_\ell\right)=\frac{1}{N}\log p\left(\mathbf{k}_\ell,W_\ell\right)$ which is determined once I choose the Bayesian prior parameter likelihood ...
0
votes
1answer
91 views

Estimating the total attendance

Suppose you do not know how many people are attending a convention, but you do know that as each person entered he was given an identification tag with a number on it. The tags are numbered serially ...
0
votes
1answer
29 views

Why would a statistician or mathematician want to find the ratio between two maximum likelihood in a likelihood-ratio test?

Why would a statistician or mathematician want to find the ratio between two maximum likelihood function in a likelihood-ratio test? I know maximum likelihood is the maximum of the probability ...
1
vote
1answer
57 views

Is it possible to “customize” the multinomial distribution to your specifications?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...