# Tagged Questions

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

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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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### 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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### 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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### Identifying joint distribution

Let $Y_1$ and $Y_2$ be independent random variables with $Y_1\sim N(1,3)$ and $Y_2 \sim N(2,5).$ If $W_1=Y_1+2Y_2$ and $W_2=4Y_1-Y_2$ what is the joint distribution of $W_1$ and $W_2$? ...
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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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### Confidence interval of the parameter of $\exp$ and normal distribution from MLE?

I have a sample $X_1,X_2,\ldots,X_n$ If the sample is from exponential distribution, I want to use MLE to estimate the parameter $\beta$. I know the result that ...
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### Chi-square test of independence: show that sum of squared standard normals has chi-square distribution

I'm studying the chi-square test of independence. According to my understanding, we first hypothesize independence between variables and consider them as being normally distributed. Then we go on to ...
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### Likelihood Function for the Uniform Density $(\theta, \theta+1)$

Let the random variable X have a uniform density given by $f(x;\theta)$~$R(\theta,\theta+1)$ What is the maximum likelihood function according to the samples $X_1,\ldots,X_n$? The question is much ...
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### Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
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### 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 ...
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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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### 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 ...
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### Distribution of likelihood ratio in a test on the unknown variance of a normal sample

EDIT: I have followed up to this discussion with a second question: Hypothesis test on variance of normal sample I am preparing for a stat exam and I was trying to derive the distribution of the ...
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### Non-Linear Model Transformation

I want to transform this Non-Linear Model $y= 8-ae^{bx}$ to Linear.And my issue is in this step $lny=ln(8-ae^{bx})$ how can simplify it to reach in a linear model which is like this $y*=b0+b1x$ ...
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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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### I am running a series of experiments that I expect to have similar outcomes. What is the best method to measure statistical significance?

Following on from this comment on an answer to my previous question, I'd like to know two things: what the best statistical test I can use to measure significance on the experiments I'm running? ...
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### Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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### Defining bias function for n trial

Let a point estimate for the sample variance be given as $\hat{\sigma}^2 = \frac{1}{n}\sum\limits_{i=1}^n(X_i- \bar{X})^2$ where $n$ is the number of samples. What is the bias in this estimate as a ...
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### Bayes estimator from a geometric distribution with a uniform prior

X is a random variable with Ber(p), 0 Y is the number of trials until a success occurs. Assume the prior p is unif(0,1). I have trouble in figuring out the posterior density f(p|Y). With the ...