# Tagged Questions

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### What is the probability that the lot will NOT be passed by the Inspector? [on hold]

Batteries for torch lights are packed in boxes of 10 and a lot contains 10 boxes. A quality inspector randomly chooses a box and then checks two batteries selected randomly without replacement from ...
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### How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
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### Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2$ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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### Probability density function transformation

Probability density function of f is given as a uniform distribution, f(x)=1 and I need to find the probability distribution function of Y=X-X^2. What I have done so far is that I found the inverse ...
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### Joint Probability of Random Variables

Suppose I took measurements $\{X_i\}$, which are all independent and they follow a normal distribution $X_i\sim N(\mu,\sigma)$. I am asked for the joint probability of all of the measurements. Based ...
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### Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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### Statistics / Probability / Population Proportion / Probability of Successive Positive Outcomes

I am not really sure where to start. "While researching fish populations in a specific lake in Arkansas, it is noticed that only 1 out of 9 bass is caught after another bass (implying no other types ...
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### Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
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### Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
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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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I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
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### Order of growth in uniform distribution

Consider an i.i.d. sample $\{X_1, \ldots , X_n\}$ from the uniform distribution on $[ 0,\theta]$ and the estimator $$M_n = \max\{X_1,X_2,\ldots,X_n\}$$ What does the above statement mean? I ...
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### Finding test of critical region for sum/variance of normal distributions

Let $Y_1,....,Y_n$ denote independent, identically distributed random variables such that $Y_1$ has a normal distribution with mean $\theta$ and standard deviations $\theta$, where $\theta$ > 0. ...
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### 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 ...
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### 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 ...
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### Doubts on Bayes hypothesis test

I meet one problem on hypothesis testing in statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$. $H_i$, $i=1,2$, is the hypothesis that $T$ is from the statistical ...
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### The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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### probability question needs some help

suppose $X$ and $Y$ are independent and identically distributed random variables that are uniformly distributed on $[0,1]$ What is the PDF of $W=Y-X$ i tried to draw a picture to illustrated it ...
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### just a manipulation problem for statistics

I am a little bit confused when i was asked to Suppose $X_1, X_2,\ldots,X_n$ is a simple random sample from a continuous distribution with density function $f(x)$. Consider the new random variable ...
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### Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
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### What is the pivotal quantity

I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated. a)$X_1,....X_n$ are uniform iid on the interval ...
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### Calculate probability of total population from sample

I have a population of infinity size. Each Unit is Boolean(True/False). I take a random sample of size N of which M elements are True(and N-M are False). I'd like to know what the probability is that ...
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### 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 $\theta$?
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### 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 ...