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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Moments based on a Bootstrapped Mean

Take $X_1,\ldots,X_n$ to be distinct observations, and let $X_1^*,\ldots,X_n^*$ be a bootstrapped sample (given a uniform resampling of cases with replacement) and let $\bar{X_n}^{*} = \frac{1}{n} \...
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Bias of a Particular Estimator

Take $X_1,\ldots,X_n \sim \mathrm{Gamma}(0,\theta)$ And take $\hat{\theta} = \max\{X_1,\ldots,X_n\}$. I want to try to find the bias of this estimator. I can see that $\operatorname{bias}(\hat{\theta})...
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38 views

Finding the distribution of a minimum of a set of variables

I have been working on a problem where $X_1,...,X_n \sim \mathrm{Exp}(\beta)$ are i.i.d. and $Y = \min\left\{X_1^{0},X_2^{X_1},X_3^{X_1+X_2},...,X_n^{\sum_{i=1}^{n-1} X_i}\right\}$ . I want to find ...
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8 views

How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
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9 views

Using coefficient of variation vs standard deviation to understand trial-to-trial variability

The basic model I have is as follows: I'm studying the response time of a neuron that's affected by two sources of noise: intrinsic noise and synaptic noise (i.e. random fluctuations in the voltage ...
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42 views

What is the expected number of throws to get all six faces of dice n times [duplicate]

I want to calculate what is the expected number of throws to get all six faces of a dice n times? (for example expected number of throws to get all six faces 2/3/4 time) ? I already know the negative ...
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29 views

Calculating a three sigma limit on data

So I have a list of data points each with their own error (eg. 3 +/- 0.2 units) and I want to determine which of the data points are within 3 sigma. To do this do I just calculate the standard ...
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17 views

Improving Probability of Event='1' in Logistic Regression (SAS)

I will try to give as much background as I can and if more is needed I will gladly give more. I'm working on trying to find an equation that will tell what probability you have a defaulting on a loan ...
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14 views

Order statistics question

In paper "Gene prioritization through genomic data fusion", they used order statistics to get combined rank for features. enter image description here ri is the rank ratio for data source i, N is ...
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13 views

Weighted Linear Regression

I am performing linear regression analysis on a time-series of data. Data contains some missing values, My question is, if I impute the missing values using mean of all the values and I want to ...
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31 views

How to know when to use t-value or z-value?

I'm doing 2 statistics exercises: The 1st: An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If ...
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28 views

Minimize the sum of distances between a sample and two “centers”

Suppose we have a set of readings $\{X_{i}\}$, each of which is a real number. What I want is to find 2 numbers, $a$ and $b$, such that minimize the sum of distances between each $X_{i}$ and ...
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2answers
51 views

Proof that: $I-X(X^TX)^{-1}X^T=((I-X(X^TX)^{-1}X^T)^2$

I have to verify if the statment: $$I-X(X^TX)^{-1}X^T=((I-X(X^TX)^{-1}X^T)^2$$ is true or not. (I do not get to demonstrate). Any help or hint will be very useful for me. Thanks a lot!
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66 views

Coin-toss game with \$1 entry fee and \$3 payout [closed]

Imagine a coin-tossing bet game. You pay \$1 to play the game (for one toss), and if you win you get a prize of \$3. The \$1 to play is not refunded. The probability of winning and losing is equal. If ...
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32 views

How do I compare rates of error between two different sample sizes?

I'm unsure on how to normalize for two different variables. Person A makes 20 pastries total, whereas Person B makes 50. 5 of those pastries, so 25%, are sampled from Person A; 10 for Person B, for ...
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24 views

Variance of exponential moving average

I'm not quite sure what the form for the rolling recursive variance should look like. I have that the recursive average is $m_t = \alpha x_t +(1-a)m_{t-1}$. Then would the rolling recursive variance ...
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31 views

Which Statistical method should I use? [migrated]

I have been given a problem and need to find the best statistical method that would help. The problem is as follows: A business has approx. 100,000 active customers who spend between £500 per annum ...
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1answer
35 views

Show that $n \to \infty, \sqrt{n}(Y_{n}-p) \rightarrow N(0,p(1-p)$

Can anyone show me the correct working out to find the variance for $Y_n$, My variance seems to be $\frac{p(1-p)}{n}$
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21 views

What test to use? [closed]

For my AP Statistics final, my teacher had our class do a research project where we collect data of our own intetesting subject and then present our findings. I did a response semi-survey thing using ...
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29 views

Question about Moment generating functions, precision, rounding errors

Hello I have a question in regard to moment generating functions and something I noticed but wasnt to clear on. Say that a random variable $X$ has $$MGF=M_{X}(t)=\frac{4}{4-t^2}$$ for $-2 \lt t \lt 2$...
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1answer
18 views

I am getting asteriks in minitab for the p--value when i try to do a Two-Way ANOVA

I am trying to do a Two-Way ANOVA. I did what i think I need to do to have the p-value show up on the screen (after i click the button that starts the calculation) but, all i see next to the P-value ...
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54 views

Distribution of the ratio of two generalized gamma distribution with same location parameters?

X $\sim$ GG(p,d,$\theta_{1}$,$\mu$) where p is power, d is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider Y $\sim$ GG(p,d,$\theta_{2}$,$\mu$) where p is power, d is shape, $\...
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1answer
35 views

Bivariate normal distributed vector $X (X,Y)$. Show distribution of $(X-Y, X+Y)$.

I have a a bivariate normally distributed random vector $X = (X,Y)$ and with Expected Value $(X)= (\mu(x),\mu(y))$, and Covariance Matrix $2\times 2$. (not independent) Now I want to show which ...
2
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1answer
46 views

Determining if dice are fair using statistics

A casino must demonstrate to the state gaming commission that every die it uses is fair. a machine is used to roll each die 1200 times. a die is discarded if the following assumption it is fair is ...
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17 views

Updating Bernoulli probability

I would like to show that the expression $ \frac{E\,\left[\, p^{t+1}\left(1-p\right)^{\left(n-t\right)}\right]}{E\left[\, p^{t}\left(1-p\right)^{\left(n-t\right)}\right]} $ , where $p$ is random on $[...
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1answer
13 views

Finding a unique point in a histogram that can be determined even if the histogram is circularly shifted.

Circularly shifted histogram: Modular addition (or subtraction) of a constant value to all data points. Maxima, Minima, etc. might not be a unique value, therefore can't be used. Edit: Let me ...
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2answers
27 views

Sigma Sum Properties

My Probability and Statistics teacher wrote the following: http://imgur.com/z0rgylh How can I prove such step? Thank you very much!
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1answer
46 views

Probability of basketball shots in a row

I've read some of the answer with this related question but it's not somewhat the answer I'm looking for. A player has a $75\%$ free throw percentage. Can someone explain to me why the probability of ...
0
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1answer
25 views

Distribution table for number of white balls without replacement

I have a box with $4$ white and $2$ black balls. I want to find the random variable distribution table for the number of white balls when I take $3$ balls without replacement. Now if we had ...
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26 views

coefficient of determination: absence of cross products [closed]

With regard to the coefficient of determination, why is the total variation equal to the sum of the explained variation and the unexplained variation and there are no cross-products?
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161 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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1answer
34 views

Finding the confidence interval of a normally distributed sample

Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of 27 vehicles was 91.29 km/h, with the sample standard deviation being 4.94 km/h. We will ...
2
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1answer
30 views

Probability of Detecting an Event Given No Previous Detections

I'm trying to figure out: Given a probability distribution of how likely an event is to be detected at time $t=1,t=2,t=3,t=4$ etc, what is the probability that it will be detected at t=n+1 if it is ...
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19 views

Importance sampling example

I have a question regarding importance sampling. During a lecture we were told that importance sampling is simply a shift to another PDF to improve point sampling. \begin{equation} I=\int_{a}^{b}dx\ f(...
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1answer
26 views

Do we not have that $X_1, \ldots, X_n \sim F$ iff $X_1, \ldots, X_n \sim f$?

From pg. 39 of All of Statistics: If $X_1, \ldots, X_n$ are independent and each has the same marginal distribution CDF $F$, we say that $X_1, \ldots, X_n$ are IID (independent and identically ...
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1answer
25 views

Best fit with unknown functional form

I have data from a simulation of a certain function $F(x)$ for a discrete set of values $\{x_i\}$. I know that $$F(x) = A \ f(x) + B \ g(x)$$ where $A,B$ are (real, positive) constants and $f,g$ ...
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12 views

Comparing two observations with given significance level

My data is as follows: for each of French municipalities the mean income of inhabitants is given. My sets consists therefore of pairs (municipality, mean income). I am asked to compare two ...
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2answers
20 views

Integration Question Arising from Attempting to Compute a Marginal Distribution

I have a question about the last step of a proof on page 37 of All of Statistics. The entire proof is here for sake of completeness, but I don't think grokking it in its entirety is necessary for ...
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2answers
22 views

Trouble with a Probability (Integral) Computation from *All of Statistics*

From All of Statistics pg. 36: Suppose that $X$ and $Y$ are independent and both have the same density $f(x) = 2x$ for $0 \le x \le 1$ and $f(x) = 0$ otherwise. Let us find $\mathbb{P}(X + Y \le 1)...
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49 views

Rank a pool of $N$ items with validation & weight

I'm not a mathematician and I'm new to this site. Apologies if this question is not formulated correctly. I have a statistical problem I need to resolve. Any help is highly appreciated! I have $N$ ...
3
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1answer
29 views

Coherent configurations (example and explanation)

A coherent configuration is a pair $(X, S)$ consisting of a finite set $X $ of size $v$ and a set $S$ of binary relations on $X$ such that •$ S$ is a partition of $X × X$; • the diagonal relation $ ...
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11 views

Stuck on a step in the derivation of the variance of a sample variance

I am trying to understand the formula for the variance of a sample variance $$var(S^2_n) = \frac{1}{n} \left [ \mu_4 - \frac{n-3}{n-1}\cdot \sigma^4 \right ] $$ We start from: $$ var(S^2_n) = E(S^...
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3-parameter Weibull distribution

Given a 3-parameter Weibull distribution, how does one find the magnitude of the maximum event that will on average occur one time or more in a given time period? Example: for a Weibull ...
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1answer
18 views

Quesiton about interquartile with continous functions

I am having trouble understanding how to find the interquartile range for a function when it is defined piece wise. For example, I have a density function that is a basic triangle ie $f(x)=\frac{x}{...
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1answer
34 views

How to find value of RV for this condtion [closed]

The probability density function of the time to failure of an electronic component in a copier(in hours) is $$f(x)=\exp(-x/1000)/1000$$ for $x>0$ and $f(x)=0$ for $x\leq 0$. How determine the ...
0
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1answer
28 views

Extension of Simple Random Sample without Replacement

First, I will show an extension of simple random sample without replacement, and then put forward the question. (1) From Wiki, a simple random sample is a subset of individuals (a sample) chosen ...
2
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1answer
31 views

Central Limit Theorem: Show that $\mathrm{Pois}(n)$ distribution if approximately Normal if $n$ is a large positive integer

Here is what the solution in the textbook says: Let $S_n=X_1+...+X_n$, with $X_1,X_2,...$ i.i.d. $\sim \mathrm{Pois}(1)$. Then $S_n \sim\mathrm{Pois}(n)$ and for $n$ large, $S_n$ is approximately $...
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1answer
37 views

Create a strictly increasing sequence following criterias

Problem Let y be a sequence of real numbers (of length $n$) bounded in the range [0,1]. I am trying to calculate the sequence x ...
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42 views

A question about Chebyshev's inequality and why $\operatorname{Var}(\overline X_n) = \sigma^2/n.$

This is the question: For i.i.d. r.v.s $X_1, . . . ,X_n$ with mean $μ$ and variance $\sigma^2$, give a value of $n$ (as a specific number) that will ensure that there is at least a $99\%$ chance ...
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249 views

Which are the most fundamental theorems in core undergrad math? [closed]

I want to compile the most important theorems for Calculus, Linear Algebra, and introductory ODE, PDE, probability and statistics. These should be the most important 5/10 per "subject", to put an ...