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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Which descriptive statistics are sufficient?

I was recently asked to choose from the basic set of descriptive statistics which ones are sufficient. Mean Median Mode Standard deviation E. Variance Range IQR (or IR) Skewness ...
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53 views

Effect on change in estimated value on Percent Error

Here is how i calculate the percent of error (Estimated - Actual)/Actual * 100 where the estimated value is obtained by subtracting two different numbers; let us ...
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97 views

Closed form of chi-square divergence between two gaussians

If $f_1$ is the pdf of a $N(\mu,\sigma^2)$ and $f_2$ is the pdf of a $N(\nu,\tau^2)$, can I compute the Chi Squared divergence between $f_1$ and $f_2$ in closed form (as a function of the parameters ...
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362 views

Estimator for sum of independent and identically distributed (iid) variables

Consider the Chernoff bound described in Theorem 1 of this paper: Theorem 1. Let $X_1,\ldots,X_n$ be discrete, independent random variables such that $E[X_i] = 0$ and $|X_i|<1$ for all $i$. Let ...
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290 views

Mean and Median in a Classic River Crossing Problem

Consider the following classic problem: Four people on the west side of a river wish to use their single boat to get to the east side of a river. Each boat ride can hold at most two people, and the ...
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124 views

How can I calculate how many iterations of a benchmark test to do?

I have a test which benchmarks the performance of my 3D rendering software, in frames per second. I run the test for each revision of my rendering software to check that I'm improving performance. For ...
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1answer
404 views

How to calculate percentile rank if all values are equal to zero

The formula for finding Percentile rank as per wikipedia is One of the edge cases in my application is when all the values are zero. In this case the number of scores less than the score of ...
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1answer
45 views

How many different starting hands are there?

There is a deck with 60 cards, with 11 different types of cards. It contains 20 A cards, 4 B cards, 4 C cards, 4 D cards, 4 E cards, 4 F card, 4 G cards, 4 H cards, 4 I cards, 4 J cards and 4 K cards ...
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133 views

Which statistics are sufficient?

Can anyone help explain why the following statistics are either Sufficient or not Sufficient? Mode, Mean, Median, Standard Deviation, Skewness, Kurtosis, Range, IQR
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1answer
2k views

Is standard deviation additive?

I am helping someone study for a statistics exam. I am quite good at most other math classes but it's been a while since I studied statistics. I am stuck on one of the exercise problems that we worked ...
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30 views

Probability poser

We play a variation of 6 card brag but have a debate over the accurate ranking of the hands. 6 cards are dealt to each player. 2 hands of 3 cards must be made. Bearing in mind that each 3 card hand ...
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1answer
51 views

Probability question with Normalized Curve

This is driving me crazy... "A Machine fills a 3 pound coffee can with a measured amount of coffee. The Weight has a normal distribution with a mean of 3.1 pounds. The standard deviation is 2.0 ...
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151 views

Can a batsman's score be predicted in the sport of cricket?

I was browsing The Signal and the Noise in bookstore and chanced upon a chapter about predictions in baseball and found out about Moneyball and sabermetrics. I understand the closest to predicting a ...
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1answer
80 views

Standard error of a statistic

the standard error of a standard deviation is given as : s/n^(1/2). Would the standard error of kurtosis and skewness follow the same idea? For example, se of kurtosis = kurtosis/n^(1/2)?
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1answer
44 views

How do I determine which Single Variable Discrete Model is appropriate to use?

I am taking a statistics course and we are studying Random Variables and Probability Models right now. I am learning a lot about the probability functions for different variables, but I am having a ...
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1answer
101 views

Dependent chances using Bayes' rule

The chance that somebody get's mad cow disease is 0.01 (1%). If someone visits the USA this chance becomes 0.05 (5%). The chance that somebody goes to the USA is 0.01 (1%). If someone goes to the USA, ...
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2answers
383 views

Fitting normal distribution to the data

I have been given a set of data points. How can I find the best fit of the form $$\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{\sigma^2}}?$$ Even better if Sage can do it. And how can I approximate ...
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1answer
143 views

Compare 2 lists , t-test problem?

I have 2 lists, the first list those that bought books and the second the list readers of the same books. I want to compare them to see if there is scientific difference between them. I am planning to ...
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1answer
164 views

Do multiple polls with identical margins of error decrease the total margin of error?

Is it possible to take multiple polls and combine their margin of error to produce a single, overall margin of error? For an example, let's say we have two populations with the following ...
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1answer
87 views

Least square problems

Let the least square problem be $Ax=b$. If the vector $v$ satisfies $A^*v=0$, why does adding a multiple of $v$ to the right hand side $b$ does not change the solution, e.g., $Ax=b+tv$, for $t$ a ...
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1answer
138 views

Moment Generating Function, find a.

Hi, having touble with this moment generating function problem. I'm given the moment generating function $M_X(t)=\frac {a} {3e^{-t}-2}$ and need to find $a$ and the largest admissible domain for ...
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2answers
107 views

Given Probability Function, find $k$.

Having trouble with this question. Suppose $X$ is a random variable with probability function: $f_X(x)=k/x^2$ I need to use a "Basel Problem" to find k and prove that the expected value $E(X)$ ...
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1answer
238 views

Bacteria Posson Distribution Problem

I'm having some difficulty two questions and was wondering if you could help me out. It goes something like this: a kind of bacteria is distributed in water according to a PPP (Poisson Point Process). ...
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1answer
122 views

absolute value of state space necessarily a markov chain?

Suppose $X_t$ is a first-order Markov Chain with state space $\{-1, 0, 1\}$, and transition matrix $P$, Is $|X_t|$ (absolute value) necessarily a Markov chain? Thanks!
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1answer
13k views

How do I find the standard deviation of this if I'm only given the mean?

So I have this problem that goes like this: Suppose the distribution of height over a large population of individuals is approximately normal. Ten percent of individuals in the population are over 72 ...
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2answers
317 views

Variance of discrete random variables

Two fair and independent dice (each with six faces) are thrown. Let $X_1$ be the score on the first die and $X_2$ the score on the second. Let $X = X_1 + X_2$ , $Y = X_1 X_2$ and $Z = \min(X_1; ...
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260 views

Is there a statistical hypothesis test that uses the mode?

Is there a statistical hypothesis test that considers the mode rather than the mean or median?
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192 views

What statistical hypothesis test to use for comparing results of two equations?

Given a function $f\left(x\right)$, I have two formulas to compute the coefficients of the same harmonic series approximation to $f\left(x\right)$. Call the results of each formula $^1c_k$ and $^2c_k$ ...
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131 views

Expansion Conditional Variance

Whe know that $\mathrm{Cov}(X,Y)=E(X.Y)-E(X).E(Y)$ so if we consider this equation what happened if $X$ & $Y$ being conditional random variable $\mathrm{Cov}((X|Z_i),(Y|Z_j))=?$
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1answer
48 views

simplifying an expression involving an integral

Simplify the following expression $$ \iint_{-\infty}^{c+x}xf(x)f(y) \,dy\,dx+\iint_{c+x}^{\infty}yf(x)f(y) \,dy\,dx $$ where $x$ and $y$ are iid random variables; $c$ is a constant; and $f$ is the ...
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1answer
178 views

p-lim inf definition (limit inferior in probability)

For an arbitrary sequence of real-valued random variables $\{Z_n\}_1^\infty$ , we define limit inferior in probability as follow : $$ p-\liminf_{n\to \infty} Z_n \equiv \sup \{ \beta|\lim_{n\to ...
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1answer
84 views

Interpreting an integral/ probability

Think of two iid random variables $x$ and $y$ with density $f$ and CDF $F$ and a constant $c$. What could the qualitative meaning of the following expression be? ...
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2answers
346 views

Finding the probability that the temperature is between x and y

Hello I have a problem that I can't seem to come to the proper conclusion with. It is as follows: What is the probability that a randomly selected day in August will have a temperature greater than ...
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1answer
379 views

Regression question - Weak exogeneity

The typical assumption of linear regression, weak exogeneity, states, $$E(\bf{\epsilon_{i}})=0$$ when the regressors are fixed and $$E(\bf{\epsilon_{i}}|\bf{x_{i}})=0$$ when the regressors are random. ...
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304 views

Skewness-adjusted t-statistic

I've come across two different forms of a skewness-adjusted t-statistic, which was developed originally by Johnson (1978): $$ J = t + \frac{gt^2}{3n} + \frac{g}{6n} $$ and $$ J = t + ...
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1answer
101 views

How to differentiate a part of normal likelihood function

$e^{-1/2 \sum_{i=1}^n (x_i - \theta)^2}$ wrt to $\theta$? (Without log.)
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563 views

Conditional expectation of the product of two independent random variables

Suppose that $a$ and $b$ are independently distributed random variables, with means; $\mu_a$, $\mu_b$ and variances; $\sigma_{a}^2$, $\sigma_b^2$, respectively. Further, let $c=ab + e$, where $e$ is ...
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1answer
1k views

Estimate confidence interval for true positive rate and false positive rate

I asked a question in the statistics stack exchange about "Error of generalized classifier performance" http://stats.stackexchange.com/questions/41400/error-of-generalized-classifier-performance : ...
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2answers
186 views

Find expectation of conditional random variables

Let $X$ and $Y$ be independent exponential random variables with means 1 and 2 respectively. Let $Z = 2X + Y$. How can I find $E(X|Z)$?
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1answer
314 views

Find normal random variable from uniform

Suppose $U$ and $V$ are independent random variables with density $f(u)$ and $g(v)$ respectively. The domain of $U$ is the interval $(0, 1)$ and the domain of $V$ is $v > 0$. After the ...
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1answer
257 views

Find joint density from transformation

I'm trying to learn some tricks for how to find joint densities from transformations, but this problem looks really hard. Any help? Suppose the joint density of continuous random variables $X,Y,Z$ is ...
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1answer
56 views

Statistics and clarification of Central Limit Theorm

If I have 1,000 participants ranking on a scale of 1 to 10 regarding some object how do I interpret the confidence level and margin of error of the resulting rank? I am used to of seeing 99% ...
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1answer
656 views

Expectation of max of independent (unknown distribution) random variables.

Let's say we have 50 independent random variables $X_i$ with the same distribution. Let's also say that $E(X_i) = \mu$ and $Var(X_i) = \sigma^2 > 0$ are known values. Is it possible to determine a ...
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1answer
624 views

Sufficient statistic for the Negative-Binomial Distribution

I am fairly new to this topic but here is my problem: I have stumbled across a paper (Robinson and Smyth, 2008) stating that the sample sum is a sufficient statistic for NB-distributed random ...
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0answers
106 views

Hypergeometric distribution with a random number of draws?

I was wondering if any classical distributions can be arrived at via a hypergeometric r.v. where the number of draws from the urn is also a random variable? Ideally I would like the sample space for ...
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1answer
148 views

Find the distribution function of X.

Let the point (u, v) be chosen uniformly from the square 0<=u<=1, 0<=v<=1. Let X be the random variable that assigns to the point (u, v) the number u+v. Find the distribution function of ...
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530 views

Continuous Mapping Theorem (CMT) for a sequence of random vectors

I need help proving the Continuous Mapping Theorem (CMT) for random vectors. I'm currently reading Econometric Analysis for Cross Section and Panel Data by Jeffrey M. Wooldridge (Chapter 3, pp. 40 - ...
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1answer
111 views

Relation of mean of standard deviations and standard deviation

Let $\{x_{i,j} : i=1..7,j=1,..n\}$ be a set of samples from $n$ weeks (where $i$ denotes the day of the week). Is there any interesting information to be gleaned from the relationship (ratio, ...
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1answer
50 views

What's the distribution of a row in a normal distributed data?

I'm working my assignment and it's a really odd assignment to me. For:- Show $\Sigma$ given by (1) is qualified as a covariance matrix. I don't know if there is a qualification criteria for a ...
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4answers
2k views

iid variables, do they need to have the same mean and variance?

If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different ...