# 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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### Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
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### How many time the standard deviation, do I need to travel from mean in both directions such that I cover a given percentage of data?

I do not have much experience in Statistics. However, I read this rule on a page and followed it up on Wikipedia: https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule I wanted to know ...
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### Show that $(\bar{X})^2$ is not an unbiased estimator for $\mu^2$

If $X_1, ... , X_n$ are n identical distributed independent random variables each with mean $\mu$ and variance 1. A little confused by this question. Is it asking for if $(\bar{X})^2$ != $\mu^2$. ...
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### Transforming a categorical distribution by repeating trials and taking a plurality

Suppose you have a K-sided, weighted die. This is represented by a categorical distribution. Now, let's say you roll the die N times, and then pick a "winner" by choosing whichever outcome has a ...
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### Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
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### probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
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### Reference Request - Statistics Book with exercises

I'm looking for an as complete as possible statistics book with exercises, including the following topics: Probability Review Random Variables and Samples Descriptive Statistics Estimation (...
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### Two-tailed hypothesis test; Why do we multiply p-value by two?

I understand that in a two-tailed hypothesis test, we must multiply the p-value by two. i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed ...
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### Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
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### expectation and variance of an implicit estimator

Suppose the following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+...
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### Sample Standard Deviation vs. Population Standard Deviation

I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values: Sample Population I ...
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### Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
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### Probability of a slot having exactly $K$ elements

From this question asked in an interview: Consider a hash table with $M$ slots. Suppose hash value is uniformly distributed between $1$ to $M$. Suppose we put $N$ keys into this $M$-slotted ...
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### Convolution of multiple exponential distributions

I'm trying to figure out the derivation presented on page 442 of this paper. Given a probability distribution $$f_n(t) \frac{\binom{n+1}{2}}{2N}\exp{\left(-\frac{t\binom{n+1}{2}}{2N}\right)}$$ ...
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The following journal article seems to suggest Over a $5$-minute period there is a correlation between returns The average return is $0.037\%$ The average daily gain is $0.59\%$ Would anyone know ...
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### Having trouble calculating expected value?

In a mall, a survey found that the number of people who pass by JCPenney between 4:00 and 5:00 pm is a Poisson random variable with parameter λ = 100. Assume that each person may enter the store, ...
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### Need help finding joint probability density function

Let X and Z be independent random variables with X uniformly distributed on (−1, 1) and Z uniformly distributed on (0, 0.1). Let $Y = X^2 + Z$. Then X and Y are dependent. Find the joint pdf of X ...
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### Variance and Standard Deviation of multiple dice rolls

I'm trying to determine what the variance of rolling $5$ pairs of two dice are when the sums of all $5$ pairs are added up (i.e. ranging from $10$ to $60$). My first question is, when I calculate the ...
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### Conf. Interval for a simple mixture of Normals

Imagine there's a prob. p=0.5 of choosing one machine or the other to take some measurements $X$ for an experiment. One machine ($N(\theta,10)$) is much less precise than the other($N(\theta,0.1)$) ...
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### Is $2\bar x(1 - \bar x) - \sum_{i=1}^n 2 x_i (1-x_i) = 2 \sum_{i=1}^n (x_i - \bar x)^2$ true?

In Nei 1973, right after equation (9), the author says: [..] it can be shown that $H_T = 2\bar x(1 - \bar x)$ and $D_{st} = 2 \sigma_x^2$, where $\bar x$ and $\sigma_x^2$ are the mean and variance ...
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### Find the Posterior distribution- prior: $\exp(1)$, likelihood: $poisson(\lambda)($

I have a prior $\lambda \sim \exp(1)$ and a likelihood $X \sim poisson(\lambda)$, and I observed in a sample of $n=5$ a mean of $3$. What is the posterior distribution of $\lambda$? Here is my ...
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### Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
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### Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$
Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$ I tried solving this in 2 different ways. The second way is what I am stuck on: 1st Way: Let $Y=e^X$ where $X \sim N (\mu,\sigma^2)$. ...