0
votes
2answers
46 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
1
vote
1answer
13 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
0
votes
1answer
18 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
0
votes
2answers
12 views

$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
1
vote
1answer
22 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
1
vote
1answer
34 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
0
votes
0answers
22 views

Probability Distribution in Cumulative Follow-Up Study

Data layout for a cumulative type of follow-up study is : $$\text{table 01. Data layout for a cumulative follow-up study}$$ $$ \begin{array}{l|cc|l} & \text{Exposed}(E) & ...
0
votes
1answer
37 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
0
votes
1answer
28 views

A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
2
votes
1answer
39 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
1
vote
1answer
19 views

Confidence interval for difference in means

I need to obtain a 95% confidence interval for the indifference in the mean score overall I have the following data which states subject (A-L) and then Test and Retest A B C D E F G H I J K ...
0
votes
1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
3
votes
1answer
164 views

Normalization for Chi square test

The formula for the Chi-Square test statistic is the following: $\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$ where O - is observed data, and E - is expected. I'm curious why it depends on ...
0
votes
1answer
19 views

Hypothesis testing using chi-square distribbution

Four players meet weekly and play eight hands of cards. Over a year, one of the payers finds that he has won x of the eight hands with frequency fx given in the following table: x 0 1 2 3 4 5 6 7 8 ...
0
votes
1answer
48 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
0
votes
0answers
26 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
20 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
45 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
1answer
44 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 ...
1
vote
3answers
82 views

Joint pdf of $f(x,y)=e^{-(x+y)}$

I am asked to find the joint pdf of $\ f(x,y)=e^{-(x+y)} $ where $x,y$ are between $0$ and $\infty$ or $0$ otherwise. I can see its an exponential distribution, which means its continuous so I ...
1
vote
1answer
37 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 ...
0
votes
0answers
20 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 = ...
1
vote
0answers
27 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 ...
1
vote
1answer
107 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 ...
0
votes
0answers
25 views

Bayesian Variable and Model Selection, Books and Review Papers Desired

I'm hoping that the community will be able to suggest some literature for studying this topic. There seems to be very few books on the subject. There are some chapters in some books which provide ...
1
vote
1answer
52 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
19 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
1answer
42 views

proving unbiasedness of an estimator

Question given independent random variable $X_{1},X_{2},...,X_{n}$ from a geometric distribution with parameter $p$. we have an estimator for $p$, mainly $T=Y/n$ where Y is number of $i$ that ...
3
votes
1answer
224 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
46 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 ...
0
votes
2answers
83 views

Theoretical impossibility? Deviation from normality with a sample greater than 300?

Huge thanks in advance! I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The ...
1
vote
1answer
292 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 ...
0
votes
1answer
259 views

Multivariate normal distribution conditional on two random variables

I am given a dataset with in each column a set of data pertaining to a different random variable. I know that the data are normally distributed. Now how can I find the estimated mean and variance of ...
1
vote
1answer
61 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} ...
1
vote
1answer
50 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
1
vote
1answer
148 views

How to calculate probability using multinomial distribution?

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} ...
-1
votes
1answer
174 views

Jensen inequality

Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random ...
0
votes
1answer
50 views

Fitting probability distributions based on moment generating functions

Say I have a random variable $X$ with mgf $M_X(t) = 1 + a_1t + a_2t^2 + a_3t^3 + \cdots $ and another random variable $Y$ with a probability distribution determined by two parameters $\theta_1$ and ...
1
vote
1answer
64 views

An Exercise of noncentral $\chi^2$ distribution.

Let $Y_1,\ldots,Y_n$ be independent random variables with $Y_k$ distributed as $N\sim(a_k,\sigma^2)$, and $\bar Y=\sum_{k=1}^{n}\frac{Y_k}{n}$ denote the sample mean, $S^2$ denotes the sample ...
1
vote
1answer
72 views

cumulants of non-central $\chi^2$ distribution

Cumulant generating function is defined by logarithm of moment generating function. $$K_X(t)=\log M_X(t)$$ Let $X$ be a non-central $\chi^2$ variate with parameters degrees of freedom, $n$ and ...
3
votes
1answer
3k views

Maximum Likelihood Estimator for Multinomial.

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
2
votes
1answer
80 views

Likelihood Functon.

$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ . What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ ...
1
vote
1answer
38 views

help me with this regarding hypothesis using chi square distribution

The rope used in a lift produced by a certain manufacturer is known to have a mean tensile breaking strength of 1700 kg and standard deviation 10.5kg. A new component is added to the material which ...
0
votes
2answers
73 views

Test of hypothesis

Can you help for solving this question.What ı will use to solve this problem.I try to do something but ı thınk not correct.ı have an exam please help me
1
vote
0answers
24 views

What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]

Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads? How do I apply Chernoff Bounds to this? I really need help understanding Chernoff Bounds.
0
votes
0answers
45 views

What is a topic I could easily collect data on that follows a poisson distribution?

So I have a project for my stat class and we have to form a hypothesis or question that I can do probabilistic modeling on. I really want to do a topic similar to a poisson distribution but I'm a ...
2
votes
2answers
637 views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
0
votes
0answers
37 views

first order auto regressive

Hello how to do the following: Suppose that $X =(X_1,X_2,...,X_n)$ follows the following: $X_t - \mu = \eta (X_{t-1} - \mu) + \epsilon_t,$ $t= 1,2,...$ where $\mu \in R$ and $\eta \in (-1,1)$ ...
0
votes
2answers
115 views

Mean of a possible MLE ,cant figure out the distribution of the sum of $x^{2}$

I would like to know: Is $\beta$ a MLE? If yes what is the mean of it: $E(\beta)=$? given: $x$ is a random variable $$f(x)=\sqrt{\frac{2}{\pi \theta^2}}\exp ...