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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Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
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14 views

Data Significance. How do I analyse my data to find meaningful information?

I have data on the time taken and CPU ticks taken for different cryptography algorithms to run. I used multiple size files, different key sizes and obviously different algorithms over multiple ...
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36 views

How do I determine if a statistical relationship exists in a real life problem: honor roll class assignments?

My question relates to Determine if being on honor roll gives an advantage in being assigned to the math class (I've seen similar questions involving hits on a webpage) which I believe should be ...
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17 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
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19 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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23 views

Deriving a formula for a confidence interval

Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties: A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $ Another random ] ...
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29 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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16 views

F-test and T-test produce the same results

I am modelling a stochastic process by two different methodologies and I expect the results of each to be normally distributed with identical means and stdevs. To test that the distributions after ...
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16 views

Required sub-chapters or more materials needed to learn Statistical Inference other than my textbooks

My school is using new curriculum now and chapter "Statistical Inference" appears in my textbooks. Now I'm at second level of senior high school. I have two books, each of them has own sub-chapters ...
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Interval estimate to infer the population mean with a 95% confidence level

An industrial designer wants to determine the average time it takes for an adult to assemble a toy. 24 people were randomly chosen to assemble the toy and the time taken (in minutes) were as follows: ...
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12 views

Forecasting disputed transaction frequencies

Problem I would like to forecast credit card chargeback/dispute frequencies using historical dispute data I have recorded over time. The data I currently store includes: Disputed transaction date ...
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15 views

Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
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Testing $H_0 : \mu_x \neq \mu_y $, in a company that markets two brands of latex paint.

A company markets two brands of latex paint regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each ...
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1answer
49 views

Kruskal Wallis - Effect size

I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
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1answer
22 views

Reading P value from ANOVA table generated by R [duplicate]

I generated an ANOVA table in R and every boxes are in number except the P value shows "3.387e-05". What does that really mean?
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23 views

Definition of true density

I am reading a paper and it talks about true densities, I mean they talk about obtaining densities from data and later compare them with the true density I want to know how can I obtain a true density ...
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16 views

Probability of taking a random sample of 24 measurements and getting a mean of at least 103.6 of true population

A random sample of size n = 24 measurements is drawn from a normal population. The sample has a mean of 103.6 and a standard deviation of 12.5. If the true population is 100, find the probability of ...
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1answer
352 views

T table with infinity degrees of freedom

why is z table the same as a t table with infinity degrees of freedom. For example as df for the T distribution goes to infinity it becomes z ( standard normal) distribution. Is this true and why is ...
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1answer
15 views

Paired and unpaired data-Statistics/Hypothesis testing

I'm getting a bit confused about paired and unpaired data, for example in this question I don't understand how this data is paired. If it was the same steel pipes that were left uncoated first in ...
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25 views

Subpopulations of an island (Bayes theorem?)

Help appreciated here. An island with 2 regions, I and II, has 4 types of individuals: AX, AY, BX and BY, for which we know their exact total nos. Here A-B-X-Y are simply traits, e.g., A=Male, ...
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165 views

Finding the solution of Sufficient Statistics for this function

Suppose $X_1,X_2,\ldots,X_n$ be a random sample of distribution with probability density function $$f(x\mid\theta) = \theta x^{\theta-1},\quad 0\lt x \lt 1,\quad 0\lt \theta \lt \infty$$ how can i ...
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how to find $X^2_{0.95,14}$, $P(X^2_{9} \geq y) = 0.99$ in a TI-84 calculator?

Can someone please show the steps in how to find $X^2_{0.95,14}$ in a TI-84 calculator? My professor does not want us to use a statistics table, but to know how to use the calculator in quizzes and ...
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66 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
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2answers
117 views

Baysien Inference and error probablility

Let $\Theta$ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let $P(\Theta=1)=p$. Under the hypothesis $\Theta=0$, the random variable $X$ is uniformly ...
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Hypothesis Question

Consider the following hypothesis: $H_{0}:\mu\leq3000$ vs $H_{a}:\mu>3000$ A sample size $n$ must be decided so the risk of a type 1 error is at most 1%, and also so that if the value of ...
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43 views

What's in a name? (Sum of Squares)

I have always believed that in order to fully understand and appreciate the mathematical subtleties and ideas behind a certain concept is to understand the name given to it. For now, can someone ...
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26 views

Interpreting what this means in a paper - significantly different at the .05 level?

I am having a hard time interpreting what something means in a paper I'm trying to get through. If you care, this is the paper: Gender Differences in the Effect of Education on the Slope of ...
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1answer
34 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
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298 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
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How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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38 views

When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I ...
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Big O p Question about Eigenvalue of Random Matrix

Suppose $S_1, S_2, \dots$ are a sequence of random symmetric matrices in $\mathbb{R}^{d\times d}$. Suppose we know that $|\lambda_\max(S_n)| = O_p(b_n)$ and also that $|\lambda_\min(S_n)| = O_p(b_n)$ ...
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19 views

How can he assume SD of population equals to SD of sample means?

I'm referring to this KhanAcademy video: https://youtu.be/bekNKJoxYbQ?t=445. My question: How can he approximate the SD of the population to be equal to SD of the sample means? Isn't that SD of the ...
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showing that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$. [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
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Conditional Probability/Expectation in the EM algorithm

I'm doing a study in which I measure data under a random censoring process. The observed data which may be interpreted as the lifetime of a subject, is denoted by $t$, with the censoring variable $c$ ...
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14 views

calculating the risk function $max(\bar{X},2)$ under Squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I calculate the risk function $max(\bar{X},2)$ under Squared error loss function
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distribution of the indicator function of poisson

Let $X_1,\dots,X_n$ be independent random variables with poisson distribution Given indicator function $$ U_i=\left\{ \begin{aligned} 1 && X_1 \ = 0\\ 0 && X_1 >0 \end{aligned} ...
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domino's pizza claim

I just got a dominos promotional flier through the post and one of the graphics advertising 'create your own pizza' lists the various toppings and claims there are 'more combinations than people in ...
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19 views

limiting variances of iid sample mean

In the book Statistical Inference (George Casella 2nd ed.), page 470, there is an example: $\bar{X}_n$ is the mean of $n$ iid observations, and E$X=\mu$, $\operatorname{Var}X=\sigma^2$. "If we take ...
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13 views

Some true/false statements about MLE and UMVUE for a normal distribution

Let $X_1,X_2,...,X_n$ (assume $n\geq 2$) be a random sample from an $N(\mu,\sigma^2)$ population where $-\infty<\mu <\infty$ and $\sigma^2>0$ are unknown. Which of the following statements ...
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14 views

To find the distribution of the random variable based on uniform distribution

Let $X_1,X_2,...,X_n$ be iid $U(-5,5)$ random variables. Then the distribution of the random variable $Y=-2\sum\limits_{i=1}^{10}\log(|X_i|/5)$ is (A) $\chi_{10}^2$ (B) $10\chi_{2}^2$ (C) ...
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1answer
42 views

P-value - test at $5 \%$ if there is significant difference in fuel consumption between the two petrol grades?

A car owners want to investigate if gasoline consumption of his car depends on the fuel octane number. He therefore intend to "premium" and "regular" at random and computes each time the ...
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31 views

Calculating 95% confidence interval for mean for a normal population

Consider a normal population with unknown $\mu$ and variance $\sigma^2=9$. To test $H_0:\mu=0$ against $H_1:\mu\neq 0$, a random sample of size 100 is taken. Based on this sample, the test of the form ...
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313 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
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21 views

To calculate variance, given conditional distribution

Let Y be an exponential random variable with mean $\frac{1}{\theta}$, where $\theta>0$. The conditional distribution of X given Y has Poisson distribution with mean Y. Then, the variance of X is ...
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estimation of the parameters of generative process modelling second-price-auction

The generative process: There are 2 entities (A,B) entity A - is the exchange performing second-price-auction entity B - is somebody who is trying to understand the distribution-of-the-value people ...
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1answer
39 views

UMVUE for pdf $f_{\theta}(x) = \theta e^{-\theta x}, x>0$

Let $X_1,\ldots,X_n$ be a random sample from a pdf $f_{\theta}(x) = \begin{cases} \theta e^{-\theta x}, & x>0 \\ 0, & \text{otherwise} \end{cases}$, where $\theta>0$ is an unknown ...
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Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...