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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20 views

Minimizing cleansing cost [closed]

Please find attached the images of questions. Help me out if you can. THe question is simple yet i am clueless.
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1answer
84 views

3 urns, each with 4 balls. select one ball from each

Three urns are labeled $1,2,3$. Each urn contains $4$ balls labeled $1,2,3,4$. A ball is drawn from each urn such that any ball is equally likely to be drawn. The number on the ball is compared to the ...
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24 views

How to get the ratio distribution of $Z=Y/X$ in this way? [closed]

I know how to do this: $$F(z)=\Pr(Z ≤z)=\Pr(Y/X ≤ z)=\Pr(Y/X ≤ z , X < 0) + \Pr(Y/X ≤ z , X ≥ 0)$$ But, I don't know how to do this: $$F(z)=\Pr(Z ≤z)=\Pr(Y/X ≤ z)=\Pr(Y/X ≤ z , Y < 0) + \Pr(Y/X ...
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1answer
23 views

monte carlo simulation: multiple variables

I am reading about the merton model in finance. It depends on multiple distributions. I want to use monte carlo simulation but I've a little question about this. If we have a statistical model that ...
3
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1answer
3k views

Margin of error and $98\%$ confidence interval question for stats people :)

The question is (this is homework, for an online class, no teacher so at times confusing) Carl conducted an experiment to determine if there is a difference in mean body temperature between men and ...
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18 views

references on social physics [closed]

I've been very curious about social physics since the moment I read the book "Big data and social physics: The lessons from a new science" written by Alex Pentland. I also know that there are many ...
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0answers
8 views

Scatter plot predict/forecast values based on historic values

I do not know if this is the correct forum, as you guys are good at math I'll give it a shot here. Now I've been thinking about this for awhile and have not found out any statistical/mathematical ...
1
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1answer
27 views

How to estimate parameters in trigram?

A popular method of computing trigram in NLP is linear interpolation: The question is how to estimate the three linear interpolation parameters to maximzie the following expression? Any form of ...
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2answers
65 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
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0answers
7 views

How to drop variables in statistical classification analysis?

Given a set of data with variables and a training set, we can proceed classification analysis using Mahalanobis distance etc.(discriminant methods) But how do we know whether all these given ...
0
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1answer
287 views

Variance of Distribution Difference

Given a distribution A and subset of that distribution B, if we only have the mean, variance, and size of both A and B, is there a way to find the variance of A - B? If not, are there other ways to ...
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2answers
47 views

How do I evaluate $\mathbb E(X\log(X))$ if $X$ has a binomial distribution, for large $n$ values?

$X\sim\mathcal {Bin}(n,p)$ I want to evaluate $\sum\limits_{x=0}^n {^n\mathrm C_x} p^x(1-p)^{n-x}x\log(x)$. Is there any way to avoid the sum because my $n$ can be very large (around $10^6$)?
4
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1answer
51 views

$E(X^2)=E(X)=1$. Find $E(X^{100}).$

$X$ is a random variable such that $E(X^2)=E(X)=1$. Find $E(X^{100}).$ My attempt: Assuming $X$ is discrete, we have $\sum x_i\mathbb P(X=x_i) = \sum x_i^2\mathbb P(X=x_i) = 1.$ We have something ...
1
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2answers
74 views

If some of your boomerangs don't come back, how many throws will you get? [closed]

Let's say you're practicing throwing boomerangs. You're not an expert, and only 50% of the time does a boomerang return to you. So you stand out in a field with 16 boomerangs and start throwing ...
0
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1answer
42 views

How to find conditional expectation $\mathbb E(X|X<M)$

Consider a random variable $X$ following the so-called folded normal distribution. That is, $X$ has density function $$ f_X(x) = \sqrt{\frac{2}{\pi\tau}}e^{-\frac{x^2}{2\tau}}, x>0. $$ ...
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0answers
18 views

What is the name (if any) of this statistical analysis concept?

I have a list of random numbers. I sort them, and designate each one as belonging to a tier of some sort (quartiles, percentiles, or even arbitrary cutoff points by proportion) based on the ...
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1answer
14 views

Power spectral density of the system output

$w(t)$ and $z(t)$: two stationary random processes $z(t) = Pw(t)$. $P$: a stable, LTI system. How to show: $$ S_z(jw) = P(jw)S_w(jw)P(jw)^*$$ $S_z(jw)$ is the power spectral density of ...
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7 views

Generalized Linear Model

$y_i$~$\space poisson\space with\space \ mean \space \mu_{i\space }where\space \mu_{i\space }=n_ie^{\beta\ x_i}\space and\space n_{i\space }is\space kown$ I have an exam coming up shortly where ...
2
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2answers
8k views

Help with proof of expected value of gamma distribution

I am struggling with this proof of the expected value for the gamma distribution. I need help with the step indicated by the red arrow. Could someone please break it down for me. Thanks.
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1answer
14 views

Sample size for close enough approximation of population standard deviation

I am new to the field of Probability and Statistics and was wondering if there actually existed a number n as big enough sample size, that would be considered the cutoff for a close-enough ...
5
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2answers
3k views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
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3answers
3k views

Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why $R^{2}$ in the regression of $Y$ on $X =$ the sample correlation between X and Y the ...
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1answer
382 views

Minimize and Maximize the variance of x

If we have 3 p(x=i) where p1+p2+p3=1. We also know that E(X)=2. How do i find the values of p1,p2,p3 that maximize the var(x) and also those that minimize it? Do we use the same method if ...
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3answers
36 views

Probability letters

I've recently gone back to my old math textbook, and this question stumped me... 8 letters: KNKVVGA, I draw 2 right away. What's the probability of getting an A and N? Probably easy for the lot of ...
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0answers
23 views

why do I get imaginary number using Discrete Fourier transform?

I am looking for an approximation for Poisson binomial distribution: The Poisson binomial distribution is the discrete probability distribution of a sum of n independent Bernoulli trials. you can ...
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0answers
9 views

Hypothesis Question

Consider the following hypothesis: $H_{0}:\mu\leq3000$ vs $H_{a}:\mu>3000$ A sample size $n$ must be decided so the risk of a type 1 error is at most 1%, and also so that if the value of ...
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18 views

geometric standard deviation

The geometric mean is like the arithmetic mean on a log scale. with the arithmetic mean it is often useful to find the standard deviation. Can the same sort of thing be done to create a geometric ...
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3answers
2k views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
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1answer
44 views

Probability of a sample mean for a bivariate probability density function

The bivariate probability density function for two random variables $X$ and $Y$ equals the following: $f(x,y)=12x^2y^3$ for $0<x<1$ and $0<y<1$; $f(x,y)=0$ otherwise. $X$ and $Y$ are ...
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10 views

optima; value of a function

Suppose we have the following function $$Err(f) = \frac{1}{2}E|Y-f(X)| = P(Y=1,f(X)=-1) + P(Y=-1,f(X)=1),$$ where $Y, f(X) \in \{-1, 1\}$. How can find the optimal value of the above function, Err? I ...
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1answer
17 views

Checking if $Z$ is an F-distribution using change of variable technique

Two independent random variables $X_1$ and $X_2$ have the following pdf $f(x_1,x_2)= e^{-x_1-x_2}$ for $x_1, x_2>0$, and $0$, otherwise. Using the change of variable technique, determine whether ...
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0answers
21 views

calculate the expected value of the call option for the following cases

For the game Caribbean Stud Poker, compute the expected value of the call option for the following cases. I try to solve with the following There are akj there If 8 comes it will complete the ...
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0answers
14 views

how to understand 'expected max approximation error'

The background is that: E() denotes the expectation and $y$ satisfies a certain probability distribution $g(y)$, then we independently sample $y_1,y_2$ from $g(y)$. It is assumed that $E(y_1-E(y))=0, ...
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26 views

Interpreting what this means in a paper - significantly different at the .05 level?

I am having a hard time interpreting what something means in a paper I'm trying to get through. If you care, this is the paper: Gender Differences in the Effect of Education on the Slope of ...
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1answer
32 views

Calculate height of histogram bins from empirical distribution function

I have an empirical distribution function: And I need to calculate the height of each of the bins[0,1], (1,3], (3,5], (5,8], (8,11], (11,14], and (14,18]. The formula to get the height is: (# of ...
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2answers
36 views

Find the indicated probability.

Question: In a batch of 8,000 clock radios 2% are defective. A sample of 11 clocks is randomly selected without replacement from the 8,000 tested. The entire batch will be rejected if at least one of ...
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0answers
13 views

Multivariate Delta Method

If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose ...
2
votes
3answers
137 views

Standard deviation of mean of a set of numbers, which are imprecise

I have a problem which seems very simple, but for some reason I can not find out what I have to do exactly. Let's say I have a set of derived values, where each of them has an individual error: ...
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1answer
22 views

Calculate Sample standard deviation and MAD

I need to solve this problem for an arbitrary N. I'm not exactly sure how to go about this. I have formulas to calculate both standard deviation and MAD, however I'm not sure what to do with the ...
2
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2answers
1k views

What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid. It is possible to relax these assumptions. For example relaxing the independent distribution ...
2
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0answers
26 views

A necessary condition for boundedness in probability

I understand that it is straightforward to show (via Markov's inequality and standard arguments) that \begin{equation} E(X_n)=O(a_n) \end{equation} implies \begin{equation} X_n=O_P(a_n) \end{equation} ...
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1answer
32 views

Calculate specified probability

Question: Suppose that $T$ is a random variable. Given that $P(-3.3 \leq T \leq 3.3) =.775$, and that $P(T<-3.3)=P(T > 3.3)$, we are to find $P(T < -3.3)$. How do I begin to solve this? ...
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0answers
31 views

Proof Borel Sigma Algebra

Let $I \equiv \lbrace [- \infty, a[ : a\in \mathbb{R}\rbrace$. Is $\sigma(I)$ Borel's sigma algebra on $\mathbb{R}$? I'm having difficulties proving these statement. I suppose it's not the Borel's ...
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1answer
34 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
4
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2answers
400 views

Variance of a max function

Say $x_1$ and $x_2$ are normal random variables with known means and standard deviations and $C$ is a constant. If $y = \max(x_1,x_2,C)$, what is $\mathrm{Var}(y)$? Well, I forgot to tell that $x_1$ ...
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1answer
25 views

$100(1-\alpha)$% approximate confidence interval

The question is: Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with density function $f(x;\theta)=\frac{1}{\theta}$ for $0\leq x\leq\theta$ where $0<\theta$. What is a ...
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72 views

Analytical Statistics Word Problem [closed]

I got parts a and b on this recent class assignment, but c onwards are a real challenge for me. Any kind of help would be greatly appreciated. Im totally stuck as to how to approach the rest of this ...
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0answers
14 views

Estimating the Square of a Mean

Suppose I want to estimate $\theta = (\mathbb{E}[f(X)])^2$, where $f: \mathbb{R} \to \mathbb{R}$ and is Borel-measureable, and $X$ is a random variable. I'll use Monte Carlo, for which one ...
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32 views

Variable weight more than another in probability

I'm about to create a ICU pediatric indicator about died and survived patients. I have the next table about the formula that I wish create. My question is... is there any subject inside the ...
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0answers
14 views

95th percentile of the statistic

Suppose $X_1, X_2, ..., X_6$ and $Y_1, Y_2, ..., Y_6$ are independent, identically distributed normal random variables, each with mean zero and variance $\sigma^2>0$. What is the 95th percentile of ...