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Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

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Approximate distribution for sum of squares of standardized Poisson random variables

Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for $\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...
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What statistical property is absent in 2-sample hypothesis tests of 5 measurements?

Suppose we observe a random sample of five measurements: 10, 13, 15, 15, 17, from a normal distribution with unknown mean $\mu_1$ and unknown variance $\sigma_1^2$. A second random sample from another ...
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8 views

Derivation of Quasi-score function from the Quasi-likelihood function

In the text book "Multivariate Statistical Modelling Based on Generalized Linear Models" by Ludwig Fahrmeir and Gerhard Tutz, we see the following results, I'm aware of how the score function is ...
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1answer
53 views

Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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1answer
30 views

How do we deduce this fisher information relation?

Given a RS $X_{1},X_{2},\ldots,X_{n}$ whose distribution is well known (unless its parameters), how do we prove the following Fischer Information relationship \begin{align*} I_{F}(\theta) =\textbf{E}\...
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14 views

How to understand bayesian inference in the framework of deeplearning?

It is said that $p \left( \theta | y _ { 1 : N } \right) \propto _ { \theta } p \left( y _ { 1 : N } | \theta \right) p ( \theta )$. And $p \left( \theta | y _ { 1 : N } \right)$ is the posterior, $ ...
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7 views

Wilks' lambda's exact distribution when one of the parameters is 1 or 2

Citing Wikipedia, From the relations between a beta and an F-distribution, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 ...
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1answer
38 views

UMVUE of $\cos\theta$ when $X_i\sim U(0,\theta)$

$X\sim U(0,\theta)$. To find the umvue of $\cos\theta$ is it enough to find the umvue of theta and substitute for it. Umvue of $\theta$ being $(n+1)X_{(n)}/n$, is the answer $\cos (n+1)X_{(n)}/n$?
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probabilities related to a transient single-server queue

Consider an $N = 1$ server queue with arrival rate $\lambda > 0$ and service rate $\mu = 1$. If the process is transient, find $\rho{_{x0}}$ for $x ≥ 1$. My attempt: The process is transient if $\...
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14 views

Compartion of GLM models through log-likelihood, deviance and chi square

I'm studying GLM models in software R. I have a dataset with the follow distribution: age, sex, years of study (ys), road or hightway (usop), and claims. I'm adjusting my model to claimns where it is ...
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1answer
25 views

How do I find a p value using a T table with a sample size of 200?

A special study is conducted to test the hypothesis that people with glaucoma have higher blood pressure than average. In the study, 200 people with glaucoma are recruited with a mean systolic blood ...
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28 views

statistical analysis [on hold]

It is hypothesized that exam performance are affected by exam anxiety and the time spent revising the relevant content. What statistical test do I use to determine if there is a significant ...
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1answer
45 views

Approximate the critical region such that the size of the test tends to $\alpha$

Consider this question, Suppose $X_1, X_2, . . . , X_n$ is a random sample from an exponential distribution with mean $\lambda$. Assume that the observed data is available on $[X_1], . . . , [X_n]$,...
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3answers
82 views

Prove that $\frac{1}{n(n-1)}\sum_{i=1}^{n}(X_{i} - \overline{X})^{2}$ is an unbiased estimate of $\text{Var}(\overline{X})$

If $X_{1},X_{2},\ldots,X_{n}$ are independent random variables with common mean $\mu$ and variances $\sigma^{2}_{1},\sigma^{2}_{2},\ldots,\sigma^{2}_{n}$. Prove that \begin{align*} \frac{1}{n(n-1)}\...
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3answers
49 views

Iterated expectations and variances examples

Suppose we generate a random variable $X$ in the following way. First we flip a fair coin. If the coin is heads, take $X$ to have a $Unif(0,1)$ distribution. If the coin is tails, take $X$ to have a $...
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1answer
182 views

MLE of the unknown radius

Consider this question, Suppose that $(X_1, Y_1),(X_2, Y_2), . . . ,(X_n, Y_n)$ are the coordinates of $n$ points chosen independently and uniformly at random within a circle with center $(0, 0)...
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23 views

When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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1answer
26 views

Asymptotic distribution of median estimator when density doesn't exist

We know that when density(say $f$) exists at the median(say $\theta$) then the median estimator(say $\hat{\theta_n}$) has the following property $$ \sqrt n(\hat{\theta_n}-\theta) \to^d N(0,1/\{4f(\...
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13 views

Naive Bayes missclassification rate across classes

I have a dataset with income, age sex and education as categorical features. I used R to create a Naive Bayes classifier as follows: income ~ age + sex + education. I got the following a-priori and ...
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Why is degree of freedom so important? [duplicate]

As far as I'm concerned, the degree of freedom is simply the number of linear equations need to be satisfied. However, it seems closely related to the statistical deduction. For example Dividing by ...
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8 views

Kurtosis risk interpretation

There is a True or False item that is confusing me a little bit. Statement says: "Kurtosis risk (also known as 'fat tails' risk) explicitly describes the situation of having more observations at ...
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1answer
36 views

Showing estimator is biased without assuming $X^TX$ is invertible?

I would like to show that the ridge regression estimator: $$\beta^R = (X^TX+\lambda I)^{-1}X^T Y$$ is biased, where $Y \sim N(X\beta, \sigma^2 I)$. If we assume that $X^TX$ is invertible, this can ...
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2answers
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Testing a null hypothesis using positive and negative z-score; accept or reject

Theorem:: To test null hypothesis $H_0$: $p_0$=$p_{1}$ versus alternate hypothesis $H_1$: $p_0 \ne p_{1}$ at the $\alpha$ level of significance, $H_0$ should be rejected if $z$ is either $(1)\le -z_{\...
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1answer
18 views

understanding derivatives of a regression spline

I am trying to understand why regression splines are continuous at their knots Suppose I am fitting a regression spline $$ E[Y|X] = \alpha + \beta_1 x + \beta_2 (x - t)^+ $$ where $(x - t)^+ = \...
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28 views

Questions about Uniformly Powerful Test

I'm using likelihood ratio test and I got to the point where I need to find the distribution of the sum of X1^m, X2^m...Xn^m. My work so far:
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1answer
27 views

Denominator in Bayes - in the continuous case, why isn't it zero?

For a continuous random variable, the probability of any particular value is zero. Only by integrating over some range is a non-zero probability obtained. The components of the Bayes theorem are ...
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1answer
26 views

Question regarding Extreme Value Theory and finding the distribution of X(n)

Hello stats stack exchange, I have a question regarding Order Statistics and the asymptotic distribution of $X_n$ which is the rv for max($X_1$, $X_2$,...,$X_n$) where $X_i$ are from some distribution....
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0answers
29 views

Probability of picking a white ball

the problem: I have a vase with a ball in it which I know to be either white or black with equal probabilities. I throw a white ball in the vase, shake, and take out a ball without looking, which ...
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1answer
31 views

critical region of a binomial population

I have the following homework problem: The number of successes in $n$ trials is to be used to test the null hypothesis that the parameter $\theta$ of a binomial population equals 0.5 against the ...
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85 views

If $F_X(z) > F_Y (z)$ for all $z\in \mathbb{R}$ then $P(X < Y ) > 0$?

I came across this question in a review of an old exam I took. I didn't get the answer correctly then, and I'm struggling to figure the answer out now. Can anyone help me reason through this? ...
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1answer
34 views

How to prove that Normal Squared Distances follow a Chi-Square distribution?

Given a multivariate normal distribution $f(x) = \frac{1}{\sqrt{(2 \pi)^n|\Sigma|}} \times \exp\left( -\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right)$ how can I prove that $ (x-\mu)' \Sigma^{-1} (x-...
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How do I know which GLM family to use?

I have the following question: I need to specify the distribution, link function and linear predictor. I know how to do find the link function and linear predictor if I know the first but don't know ...
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2answers
83 views

Order Statistics of Poisson Distribution

I have been given the following question, Let $n ≥ 2$, and $X_1, X_2, . . . ,X_n$ be independent and identically distributed $Poisson (λ)$ random variables for some $λ > 0$. Let $X_{(1)} ≤ ...
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2answers
124 views

How do we build a confidence interval for the parameter of the exponential distribution?

EDIT Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Exp}(\theta)$, where $\theta$ is not known. Precisely, $f(x|\theta) = (1/\theta)\exp(-x/\theta)$ Describe ...
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1answer
62 views

Given $X\sim\mathcal{N}(0,\sigma^{2})$, obtain the Fischer information of $\sigma$ and $\sigma^{2}$

Suppose the random variable $X\sim\mathcal{N}(0,\sigma^{2})$, where we do not know the value of the standard deviation $\sigma$. Then obtain the Fisher information $I_{F}(\sigma)$ through $X$. Suppose ...
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Mean and Variance of weighted sum of n random variables? [duplicate]

Suppose we have n jointly distributed random variables $x_i,i=1,...,n,$ with mean and variance $E(x_i)=\mu_i$, $Var(x_i)=\sigma^2_i$ and covariance $Cov(x_i,x_j)=\sigma_{ij}.$ Then the weighted sum of ...
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1answer
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Does the UMP exist?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. $$X=(X_1, X_2, X_3,\ldots, X_n)$$ and $g(λ)=\lambda(1 - e^{-λ})$ Is there a UMP (...
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1answer
27 views

Build an approximated confidence interval for $\sigma$ based on its maximum likelihood estimator

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\mathcal{N}(0,\sigma^{2})$. Build an approximated confidence interval for $\sigma$ based on its maximum likelihood ...
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1answer
45 views

Compute conjugate prior from the sample distribution

I feel like this question might be marked as duplicate because I see many similar incurring in that fate but I'll try anyway. I would say I did not find anything similar. I have been thought a ...
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2answers
27 views

Can we replace the t-Student distribution by the Normal distribution in this context?

As far as I have studied, given a normal random sample, we can build the confidence interval of the mean $\mu$ if we know the variance through the relation \begin{align*} \frac{\sqrt{n}(\overline{X}-\...
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How to design an experiment to test the effect of a drug on subjective energy level

I am currently running an experiment to test the effect of a drug (ALCAR) on my energy level (measured using a subjective 5-point scale). My goal is to determine whether or not consuming this drug ...
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1answer
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What is the distribution of a sum of binomial distributions with the same parameter q but with the sample sizes following a Poisson distribution?

Let $\{a_1,a_2,\ldots,,a_n\}$ be a random sample of a Poisson distribution. Consider the following random variables $X_1=\mathrm{Binomial}(a_1,q), ~X_2=\mathrm{Binomial}(a_2,q),\ldots,~X_n=\mathrm{...
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How do we prove this identity related to expectation and variance?

Prove that if $\textbf{a}$ is a vector of constants with the same dimension as the random vector $\textbf{X}$, then \begin{align*} \textbf{E}[(\textbf{X} - \textbf{a})(\textbf{X} - \textbf{a})^{\prime}...
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T-test appropriate to compare data sets from equipment modifications?

a bit of a real world example here. I work with a piece of equipment that has 3 settings: A, B and C. I want to determine whether the active setting has an effect on data measurements. I am wondering ...
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Likelihood ratio test of log normal distribution

$X_{1},X_{2}, … , X_{n}$ be a random sample from a $𝑁(\theta, 1)$ distribution. Instead of observing $X_{1},X_{2}, … , X_{n}$, $Y_{1},Y_{2}, … , Y_{n}$ was observed where $Y_{𝑖}= 𝑒^{X_{i}}$. Find ...
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2answers
36 views

Upper bound of normal cdf

Random variable $X\sim N(0,1)$. Show that, $P(X\geq c) \leq e^{-ct+ \frac{t^{2}}{2}}$ for $c>0$ and for all $t$ in $R$. I found that $P(X\geq c) = \Phi(-c)$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)...
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1answer
53 views

Binomial distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be random sample from $X\sim\text{Binomial}(2,\theta)$. (a) Find the least variance from the set of all unbiased estimators of $\theta$. (b) Find a sufficient ...
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18 views

Convergence in probability (asymptotic notation) result

Let $h=h_n$ be a sequence of numbers such that $h_n \rightarrow 0$ as $n \rightarrow \infty$, $\mu$ be a real constant and $f$ be some probability density function. I was wondering if the following ...
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1answer
48 views

Normal distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample from $X\sim\mathcal{N}(0,\sigma^{2})$. (a) Find the least variance from the set of all unbiased estimators of $\sigma^{2}$. (b) Find a sufficient ...
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4answers
499 views

What must someone know in statistics and machine learning? [closed]

There seem to be two different worlds in statistics. On one hand, there are the practitioners which run the same tests again and again. On the other hand, there is this overwhelming and seemingly ...