# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\quad x_1,x_2,\ldots,x_n$ ...
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### 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 ...
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### 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
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### 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.
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### 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 ...
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### 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 ...
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 $\eta \in (-1,1)$ ...
### 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 ...