Questions tagged [statistical-inference]

The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

121 questions
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Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
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Spectral densities of finite dimensional sample covariance matrices

Let $N \ge 2$ and $T > N$ be integers. In multivariate statistics it is of interest to analyze spectra of sample covariance matrices. The resolvent ${\mathfrak g}_M(z)$ encapsulates the whole ...
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Likelihood Function for the Uniform Density.

Let the random variable $X$ have a uniform density given by $$f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]}$$ where $-\infty\leq\theta\leq\infty$ the likelihood function for a sample of ...
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Justifying the Normal Approx to the Binomial Distribution through MGFs

Would absolutely love if someone could help me with this question, in a step by step way to help those who are uninitiated to Statistics and Mathematics. So, I am trying to "prove/justify" through ...
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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 $\sigma = 0.6$ ...
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Maximum Likelihood Estimator of parameters of multinomial distribution

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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Probability vs Confidence

My notes on confidence give this question: An investigator is interested in the amount of time internet users spend watching TV a week. He assumes $\sigma = 3.5$ hours and samples $n=50$ users and ...
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How to determine the number of coin tosses to identify one biased coin from another?

If coin $X$ and coin $Y$ are biased, and have the probability of turning up heads at $p$ and $q$ respectively, then given one of these coins at random, how many times must coin A be flipped in order ...
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Finding an unbiased estimator of $e^{-2\lambda}$ for Poisson distribution

If $X_1,X_2,\ldots,X_n\sim \mathrm{Pois}(\lambda)$, find an unbiased estimator of $e^{-2\lambda}$. I am actually supposed to find the UMVUE of $e^{-2\lambda}$ but I first have to find its unbiased ...
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Minimal sufficient statistic of $\operatorname{Uniform}(-\theta,\theta)$

I am seeking clarification on why both the vector $(X_{(1)},X_{(n)})^T$ and $\max\{-X_{(1)},X_{(n)}\}$ are sufficient for $\operatorname{Unif}(-\theta,\theta)$, but only $\max\{-X_{(1)},X_{(n)}\}$ is ...
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Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. ...
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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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Checking the consistency and Bias of $\frac{\sum X_i +\sqrt{n}/2}{n+\sqrt{n}}$

Let $X_1,\ldots,X_n$ be i.i.d. $B(1,\theta)$ random variables, $0<\theta<1$. Then, as an estimator $\theta$, check if $T(X_1,\ldots,X_n)= \dfrac{\sum_{i=1}^n X_i +\sqrt{n}/2}{n+\sqrt{n}}$ is ...