Tagged Questions

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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When to use Central Limit Theorem or Cramers Theorem

In for example this paper the authors say The central limit theorem provides an estimate of the probability \begin{align} P\left( \frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} > x \right) ...
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Questions dealing with Poisson distribution technique

"A waste disposal company averages 6.5 spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of 2 or more occurring ...
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Joint multivariate normal distribution

If $\mathbf{y}$ has a multivariate normal distribution $N_n(\mathbf{a},\sigma^2\mathbf{I})$, can I say that $(\mathbf{y},\mathbf{a}^\intercal\mathbf{y})$ also has a multivariate normal distribution? ...
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25 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
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What is the correct statistical language to conclude using type II error?

Update: This question arises when I read a neuroscience paper. In the natural science community, people are generally less careful about the correctness of statistical language. I am clearly aware of ...
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50% plus or minus 13% is good or bad guessing. How was this reached?

I watched a documentary some years back and I cant find it anymore. It was either about genetics or twins (specifically identical twins). But I recall a test was done to see statistically if identical ...
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Suppose U is a uniform random variable over [0, 1] [on hold]

This is one of two HW problems that I'm positing that I have no clue how to go about. Suppose U is a uniform random variable over [0, 1]. Show that $Y = (b - a)U + a$ is uniform over [a, b] Use ...
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31 views

If $E(W) = \mu$ and $Var(W) = \sigma^2$

This is one of two HW problems that I'm positing that I have no clue how to go about. If $E(W) = \mu$ and $\operatorname{Var}(W) = \sigma^2$ show that $E\left(\frac{W - \mu}{\sigma}\right) = 0$ and ...
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Will XORing any data with random data produce a random result?

Provided you have a stream of input data and a stream of random data both in the set (0,1). The random is data truly random, that is, unpredictable by the user and ...
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1answer
24 views

Statıstıc problem

Will I use binomial distribution for this question? Can you help me please thnk you
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10 views

Equal fraction of explained variance in PCA

Let's suppose we obtained this variance decomposition after performing a PCA: My question is, what can we tell about the original data? I know the variance of each principal component is the ...
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1answer
30 views

Probability Mass fuction for scratch ticket

A lottery ticket has 4 squares, each with either a star or an X. The printer printing these tickets has a 20% chance of printing a star per individual square. If the amount paid per star is as ...
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Statistic problem

Can you help me to solve this problem pls,I have exam and I am studyıng. What wıll I use, bınomial or Other thing ? Thank you
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19 views

Problem of Statistics

Let $x$ be a variable assuming values $1,2,\ldots,k$ and let $F(1)=n,\ldots,F(n)$ be the corresponding cumulative frequencies of the 'greater than' type. Show that $$\text{Mean of }x=\frac{ ...
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21 views

Statistic binomial dist

Can you help me to solve this question pls, I consider that I Will use binomial distrıbutıon but I couldnt
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1answer
17 views

Increase variation between 2 values

How can I calculate the percentual increase variation ? June: $ 2.574.724,83 July: $ 4.041.072,22 I want also do the inverse, i.e. if I know the increase ...
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36 views

Generate random number with specific probability distribution

Well consider I have a uniform random number generator. How would I craft a function which takes as parameter this RNG, such that the distribution is following a given function? Or more to the point: ...
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Finding midpoint in class interval to calculate mean for cumulative frequency graph.

In a cumulative frequency graph (or histogram), the data is often given in class intervals. To calculate the estimated mean of the data, the formula is: (SUM m*f)/(SUM f) where m = midpoint of the ...
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58 views

Geometric vs Arithmetic returns differences

Been reading some notes that say when calculating returns, using the geometric methodology (1+returns, performing a division between the two returns and - 1 from the result) for computing returns is ...
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Survival models and differential equations

I have a question regarding survival models and differential equations. Is it possible to translate survival models ( in survival analysis) into differential equations? For example can we write the ...
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2answers
43 views

Convergence Rate of Sample Average Estimator

Let $X_1, X_2,\cdots$ be i.i.d. random variables with $E(X_1) = \mu, Var(X_1) = σ^2> 0$ and let $\bar{X}_n = {X_1 + X_2 + \cdots + X_n \over n}$ be the sample average estimator. Is there a way to ...
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22 views

About matching.

A population of $n$ people, each having $m$ "features" $f_1,f_2, \dots ,f_m$ (for instance, where they live, brand of milk they consume, annual profit, etc...). Now, let $f_1$ a basic feature. Is it ...
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Given a list of N integers, how to find out if the second derivative is positive or negative?

Let's say I have a list where N=10, such as [45,34,56,22,33,44,34,34,43,35]. I would like to know if the second derivative is positive or negative, in other words, if the rate of change of these ...
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Tailored Expectation

Let $x=[x_1,\ldots,x_m]^T$ be a random vector whose elements are drawn iid from some distribution which is parametrized by some parameter $\theta$. (i.e. $x_i\sim P_{\theta}(x_i)$). I want to know if ...
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How to model and plot non-stationary mean-valued data

What would be the best way to plot a distribution of the data whose mean is non stationary. For example: if I have a data series, say y = [100, 97.3, 95.2, 93.2, 91.1, 91.2, ... 0] which yields ...
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crazy number of divisors chart on Wikipedia - $\Omega(n)$

I am learning about factorization into primes and I learned about the sum number of divisors function (counting multiplicity). $$ \Omega(n) = \Omega \left(\prod p_i^{a_i}\right) = \sum a_i $$ The ...
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Approximate proportions of data sets

I have two data sets, $A$ and $B$. I also have proportions of an event occurring in $A$ and $B$ separately. How would I use my data to approximate the proportion of this event occurring at both $A$ ...
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Multivariate regression analyisis with grouped data. How do I 'un-group'

I am trying to determine if either of two non-numeric variables effect the percentage of people who take a specific action. There is another variable that needs to be controlled for. In real world ...
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1answer
28 views

Squence of order statistic converges to median?

Suppose that $(X_1,...,X_N)$ be $N$ iid samples from uniform distribution. Let $X_{(n)}$ be $n$-th order statistic. It sounds natural that $X_{N/2}$ converges to the median, $1/2$. (interpret $N/2$ as ...
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11 views

Conditional Quantile Function

I happen to find often the expression "The conditional quantile function $Q_y(\tau|x)$", but for some reason I cannot find anywhere a precise definition of this object. Does anyone know the precise ...
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Question regarding expectations and OLS

So I am finally getting to log-based regression stuff in econometrics. I know from linear OLS, we assumed that $$E[u_i\mid x_1,x_2,\ldots]=0$$ Where the $u$ values are random variables we do not ...
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How can I minimize the error from fitting two functions to different parts of data?

I have some data, and I expect that part of the data will fit to a power law, and part of the data to a line. Here's a made up example of some similar data: Where I've made the red part a quadratic ...
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How to define probability density function in Hilbert space??

Consider the space of random continuous functions $f:(0,1)\rightarrow\mathbb R$. Suppose we assume that this is a Hilbert space. Is there any notion of probability density function in the Hilbert ...
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Conditional Probability of components in series

I've two components arranged in series one after another. I've below information with me : ...
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24 views

Normal approximations and Binomial distributions

I am having some difficulty with the following question from my textbook. I have really been trying to understand the use of normal and binomial approximations, but I'm getting really confused. Any ...
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Show that if T ~ t_n, then T^2 ~ F_(1,n)

Show that if T ~ t_n, then T^2 ~ F_(1,n) This is the third part of a practice question. I am stuck on solving it
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chi-Squared distribution

This is a question on the practice exam and I would like help. U is Chi-squared with n degrees of freedom; V is Chi-squared with m degrees of freedom. U and V are independent. Prove that Z=U+V has a ...
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Difference between variance of an IID sampling distribution and the variance any sampling distribution?

If I have a population {1,2,4,4,9} and a sample size 2, the possible samples are: {1 2}, {1 4}, {1 4}, {1 9}, {2 4}, {2 4}, {2 9}, {4 4}, {4 9}, {4 9}. I understand that the mean of the sampling ...
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PDF of X-Y and X/Y

Let X,Y ~ Uniform(0,1) be independent. Find the PDF for X-Y and X/Y. I couldn't understand the method used in the solutions. I am guessing the first part has something to do with the convolution of X ...
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Expected Number and Variance

Practice Exam question. Not sure how to do this. The number of offspring of an organism is a discrete random variable with mean $\mu$ and variance $\sigma^2$. Each of its offspring reproduce in the ...
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Expected Number of items

Need help with this practice exam question. An item is present in a list of n items with probability p; if it is present, it is position in the list is uniformly distributed. A computer program ...
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1answer
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Statistics: Maximum Error of Estimate [on hold]

A random sample of size $25$ taken from a normally distributed population yielded a sample variance of $2.25$. Which one of the following represents the value for the maximum error of estimate for a ...
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1answer
30 views

Expected Value, elevators

A building contains two elevators, one fast and one slow. The average waiting time for the slow elevator is 3 min. and the average waiting time of the fast elevator is 1 min. If a passenger chooses ...
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Probability Between Two Normally Distributed Variables

The weight of medium-size tomatoes selected at random from a bin at the local supermarket is a normal random variable with mean $μ = 10$ ounces and standard deviation $σ = 1$ ounce. Suppose we pick ...
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7 views

Application of Statistics in Sports Pool Selection

I am looking for the best mathematical approach for my hockey pool selection based on statistics from the previous season. We are 8 participants. Each participant must select: 9 Forwards 4 ...
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1answer
98 views

How do you find variance when not given probability?

A small farmer in a developing country grows cassava and groundnuts. The amount of cassava he grows per year is a random variable, X, with a mean of 500 pounds, and a variance of 625 ...
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Time Series, AR(2) MA representation

I have a problem and I couldn't deal with it. My question is as follows: Consider a stationary AR(2) process given by Xt-1.76(Xt−1)+0.8(Xt−2)=2+at Can you help me? Actually I found the mean part by ...
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1answer
12 views

confusion in percentiles

in definition of percentile, it is said that, some value is in 30th percentile means, 30 percent of the data values are lower than the given value. But my confusion is, what if I have 1000 data ...
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Order statistics/Probability

Find $P( \widehat\theta\lt X_{(n)})$ for $n \ge 2$, where $X_1,..,X_n$ is a sample from $Unif([0,\theta])$, $\theta \gt 0$ and $\widehat\theta$ is the estimator of $\theta$ obtained by using the ...
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1answer
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Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...