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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What are different geometric interpretations for the sample variance?

I primarily work with non Euclidean data and am looking to extend concepts of 'variance' to Riemannian manifolds. I am aware of Karcher variance, but I need efficient ways to solve for it. For ...
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102 views

How to prove that second derivative of $\log\big(\int_{-\infty}^x e^{\frac{-t^2}2} dt\big)$ is $>-1$?

Let $\Phi(x)=\int_{-\infty}^x e^{\frac{-t^2}2} dt$. How can I prove that $$\left[\frac{e^{\frac{-x^2}2}}{\Phi(x)}\right]'>-1?$$ I could prove that its $lim$ at $-\infty$ is $-1$ and at $\infty$ it ...
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490 views

Markov Chain Conditional Probability

A Markov chain has the transition probability matrix as follows. $$To$$ $$ From \begin{matrix} STATES& 0 & 1 & 2 \\ 0 & 0.6 & 0.3 & 0.1 \\ 1 ...
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1answer
318 views

Finding expected value and variance of a random variable.

Given $$X∼Bin(n,p_x)$$ and $$Y∼Bin(m,p_y)$$ and we have to find E[W] where W = 4X + 6Y for which I got $$E[W]=4np_x+6mp_y$$ by using $$E[aX+bY]=aE[X]+bE[Y]$$ for variance I'm not sure whether to do it ...
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1answer
101 views

Sufficient statistic.

If $X_1$ follows Binomial distribution with parameter $m$ and $p$ where $m$ is the number of trials and $p$ is the probability of success , that is , $X_1\sim B(m,p)$ and $X_2\sim B(n,p)$ then how can ...
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1answer
80 views

Inverse of a triangular matrix in a statistical problem

Can any one give to me idea how to solve this problem? Find the inverse of the triangular matrix T, where $ T =\left[ \begin{array}{ccc} I & J & J \\ 0 & I & J \\ 0 & 0 & I ...
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1answer
43 views

sum of rounding errors

My tax return involves 32 different numbers, each rounded to the nearest dollar and then added together. Assuming that the errors by rounding are uniformly distributed on the interval (-1/2,1/2), ...
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2answers
76 views

Find density of the smallest value in a sample

Let $X_1, X_2,..., X_n$ be a random sample from an infinite population with density function $f(x)$ and distribution function $F(x).$ Let $Y_{(1)}$ be the smallest value in the sample (the first order ...
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1answer
1k views

Consistency of sample variance $S^2$

Let $Y_1...Y_n$ be independent $N(\mu,\sigma^2)$ R.V.s. Their sample variance is: $$ S^2=\sum_{i=1}^n \frac{(Y_i- \overline Y)^2}{(n-1)} $$ Treating $S^2$ as an estimator, is the estimator ...
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1answer
343 views

Asymptotic Relative Efficiency: Poisson

I'm trying to find the asymptotic relative efficiency of a Poisson process: $$\frac{\lambda^t \exp(-\lambda)}{t!} = P(X=t).$$ When $X = t = 0$, the best unbiased estimator of $e^{-\lambda}$ is ...
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1answer
2k views

Properties of $\hat\sigma^2$ | bias and variance

I will first state the question I am trying to solve then I will demonstrate my thoughts! Let $Y_1...Y_n$ be independent normal R.V.s each with mean $\mu$ and variance $\sigma^2$. We want to estimate ...
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2answers
38 views

Partioning/Enumeration

How many ways can one distribute A) 15 Balls into 3 bags. Both bag and balls are distinct (labelled) and each bag must contain at least one ball. B) 10 balls into 3 bags. again both bag and balls ...
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1answer
3k views

MLE of bivariate normal distributoin

Suppose that $X$ ($n$ by $2$ matrix) follows a bivariate normal distribution $N(\mu,\sigma^2I)$, where $I$ is the $2\times 2$ identity matrix. How to find the maximum likelihood estimates of $\mu$ and ...
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1answer
104 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
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3answers
282 views

MLE of a uniform distribution

I have a question about the MLE of the following distribution. let $Y$ be a Uniform$(0,\theta)$ random variable, where $0<\theta<\infty$ and $\theta$ is to be estimated. The first question ...
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1answer
384 views

Finding MLE from CDF

So I have the following $CDF$ and I was wondering how will I be able to get the Maximum Likelihood Estimator since we do not have the $PDF$ to work with. The following $CDF$ is an exponential ...
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1answer
120 views

Maximum Likelihood Estimation

I am stuck on this problem and help would be greatly appreciated! I have the following PMF (a modified Poisson Distribution). \begin{align*} \frac{\lambda^x e^{-\lambda}}{x!(1 - e^{-\lambda})} ...
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28 views

Finding variance for RV=RV1(1-RV2-RV3-…)

When a AR model is fit using the R arima package, you get back the estimate and S.E. for all the coefficients and the "intercept". However, for a non-zero-mean process, the true intercept is really ...
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2answers
77 views

Smoothing linear graph

Which ways I can smooth data where are random fluctuations? This is linear graph of my data input: Big curves (about time 100 etc) are desirable. I already tried sliding window. Which way do you ...
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1answer
85 views

Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
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0answers
47 views

Delta Method and Expected Value

I would just like to check something regarding the delta method. After using the delta method to verify that, $$ \sqrt(n)(g(X_n)-g(\mu)) \sim n(0,\sigma^2 [g\prime(\mu)]^2) $$ we cannot generally ...
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1answer
118 views

Rao-Cramer lower bound

Find the Rao-Cramer lower bound if the random sample $X_1,X_2,...,X_n$ is taken from the distribution with the p.d.f. $$f(x;\theta)=\frac{1}{\theta}x^{\frac{1-\theta}{\theta}}$$ where $0<x<1$ ...
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1answer
116 views

Variance of Coefficients in a Simple Linear Regression

I have a linear regression model $\hat{y_i}=\hat{\beta_0}+\hat{\beta_1}x_i+\hat{\epsilon_i}$, where $\hat{\beta_0}$ and $\hat{\beta_1}$ are normally distributed unbiased estimators, and ...
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1answer
222 views

Sufficient statistic and conditional distribution intuition?

I am confused about the intuition behind the definition of a sufficient statistic. The part definition of a sufficient statistic that I am confused about is why the conditional distribution of a ...
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3answers
190 views

Health Insurance question

An insurance company issued health insurance policies to individuals. The company determined that Y, the number of claims filed by an insured in a year, is a random variable with the following ...
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1answer
140 views

Stein's Method and Coupling of random variables

Suppose a particle starts at position 5 on a number line and at each period the particle moves one position to the right with probability p and, if the particle is above position 0, moves one position ...
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1answer
100 views

Use z confidence interval to estimate population proportion

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion? (A) The sample is a random sample from the ...
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1answer
51 views

Probability that two balls have different colours

An urn contains u blue balls and w black balls, with u$\geq$w. A sample of two balls is selected at random from the urn. Prove that the probability that the balls which were picked up to have ...
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1answer
59 views

Player statistics as estimate of surreal number of game

This is a rather complex question; it may require nontrivial assumptions about human cognition. But, I'm interested in getting mathematicians' perspective. With some finagling, you can associate many ...
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1answer
41 views

Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
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2answers
85 views

Joint Moment Generating Function Help

I've been working on this problem for a while and need some direction. $$ f(x,y) = \left\{ \begin{array}{lr} \frac{1}{\sqrt{2\pi}} e^{-x} e^{-\frac{(y-x)^2}{2}} & x \geq 0, -\infty < y ...
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1answer
35 views

Probabilty Mass Function. Function which depends on past outcomes $X$

I randomly draw numbers according to the probability mass function (PMF) $X$ in which all negative values have probability zero. Each value that is drawn from $X$ can be thought of the lifespan of one ...
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1answer
702 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
39 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
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1answer
330 views

Hypergeometric Distribution : probability that more than half is good

To simplify the context, let's say that 34 % of people are ugly. haha... lets take a sample of 15 people. (n = 15) a) What is the probability that 3 or less out of the 15 are ugly ? I went on and ...
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1answer
47 views

How to show that the method to create two correlated random numbers is correct?

I would like to understand how I can show that the method to create two normally distributed random numbers given as an answer to this question is correct. Given independent $X_1$ and $X_2$ normally ...
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2answers
100 views

Maximum of two skewed normal distributions

Does there exist a means to approximate the maximum of two skewed normal distributions in terms of another skewed normal distribution? To make it clearer, given 2 skewed normal distributions ...
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3answers
3k views

Calculate the probability of an event occurring AT LEAST x times over n trials?

Forgive me if this is simple, but I've been twisting around this problem for a bit. I know how to calculate if a given event happens exactly $x$ times over $n$ trials (where $p$ is the probability of ...
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2answers
1k views

Statistics: how does standard deviation measure quality - please answer everyone?

There are many types of measures of variation/dispersion from the mean, but standard deviation can be confusing. If we are measuring error/variability, then 1σ is better than 2σ. That is to say that ...
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2answers
167 views

2D data fitting

I have some numbers as a function of 2 variables: $(x, y) \mapsto z$. I would like to know which function $z=z(x,y)$ best fits my data. Unfortunately, I don't have any hint, I mean, there's no ...
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0answers
42 views

Conditional expectation of X given Y [duplicate]

Consider the random experiment in which three fair dice are rolled simultaneously (and independently). Let X be the random variable defined as the sum of the values of these three dice. Let $Y_1$ be ...
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1answer
1k views

Is there a scientific way to assign weights to historical data?

This is a common question I usually face while processing historical data. I have year on year data of an event for the past N years.I would like to assign weights to the data of these N years so that ...
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1answer
104 views

Two Top Economist getting 9/10 correct

Suppose that we now have twenty economists instead of just one, each of whom makes their predictions based on the toss of a fair coin. what is the probability that the second most successful of the ...
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5answers
213 views

Why the natural log is there in MLE?

Why do we use natural log for MLE?
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2answers
738 views

connection between PCA and linear regression

Is there a formal link between linear regression and PCA? The goal of PCA is to decompose a matrix into a linear combination of variables that contain most of the information in the matrix. Suppose ...
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1answer
45 views

How do you describe a CDF in terms of another CDF?

This is homework, but I'm more interested in understanding the problem than the solution, so answering with a different example is totally fine. The problem has multiple parts, but I'm only stuck on ...
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1answer
172 views

Is there a convenient formula for variance of discrete random variable in terms of CDF

For a discrete random variable X, $\Omega_X\subseteq \{0,1,2,\ldots\}$, we can write $$\mathrm{E}[X] = \sum_{x=0}^\infty (1-F(x)) $$ where $F(x)$ is the cumulative distribution function of $X$. ...
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1answer
59 views

Generating a uniform distribution in the volume of a box

Suppose I have a three dimensional box, of volume $V$, and with lengths $x, y$, and $z$. I then change the box volume by $\Delta V$, such that $(V + \Delta V) = (x + \Delta x)(y + \Delta y)(z + ...
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2answers
191 views

Two-tailed or one-tailed test?

The study using 139 students studies the maximum amount of alcohol consumed. Based on data are there differences between males and females in the maximum amount of alcohol consumed? (Use alpha= .05 ...
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1answer
29 views

Meaning of “Percent increase”

When someone uses the phrase "percent increase" what does that mean? For example, if something took 4 seconds before and now it takes 1 second, would that be a 400% increase?