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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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 ...
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1answer
27 views

Probability, binomial?

I'm working on a programming assignment and am completely stuck on a problem involving statistics :/. In my problem, I am dealing with a hypothetical situation where I am walking outdoors and am ...
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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 ...
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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 ...
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1answer
28 views

Percentiles and Salary Surveys

What formula would I use (using Excel, preferably) to determine the 'average' 10th percentile rank value if I have four different and yet related 10th percentile values? Can I just average the values ...
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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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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 $10^6$)?
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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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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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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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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 ...
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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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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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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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35 views

How do I calculate PERT stddev if I use custom weight factors? [migrated]

According to (for example) http://tynerblain.com/blog/2006/04/13/foundation-series-basic-pert-estimate-tutorial/, the PERT mean estimate is $$meanA=optimistic+(4*likely)+pessimistic)/6$$ I ...
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1answer
45 views

How to find the cdf of $U=y-x$

The random variables x and y have the probability density functions $f_{X,Y}(x,y)=4e^{-2(x+y)}$. How do I find the probability density functions of $U=y-x I have tried solve by convolution theorem ...
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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 ...
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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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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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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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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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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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0answers
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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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 ...
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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 ...
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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
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 ...
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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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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 ...
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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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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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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 ...
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1answer
19 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...
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1answer
22 views

Joint normal random variables, covariance, and probability

I'm having a lot of trouble with this question: X and Y are joint normal random variables with common mean 0, common variance 1, and covariance 1/2. What is $P(X+Y\leq \sqrt{3})$? Thank you!
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30 views

Permutations of k types for a number smaller then total of k's (without replacement)

How can you find the number of permutations of a set of items grouped in different categories when you must choose less than the total of the set. For example, I have a set of $12$ songs that are in ...
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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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2answers
37 views

Find the Indicated Probability

Question: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during the ...
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1answer
16 views

Find the Variance

Question: On a multiple choice test with 9 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the variance for the number of ...
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2answers
16 views

Do you think this transition Matrix is correct?

Here is the situation we are trying to model: given a car that has 3 states, labeled 1, 2 and 3. state 1: is when the vehicle is in good operating condition. state 2: repairs may be required to ...
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1answer
28 views

Question regarding the conditional probability of multiple events

I'm having trouble really understanding a seemingly easy question about conditional probability. Here is the question: Let $n_{R}$ denote the number of red balls in an urn and $N$ denote the number ...
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1answer
24 views

Find the Indicated Mean

Question: A certain rare form of cancer occurs in 37 children in a million, so its probability is 0.000037. In the city of Normalville there are 74,090,000 children. A Poisson distribution will be ...
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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
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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1answer
37 views

Find the indicated probability?

Question: A spinner has equal regions numbered 1 through 15. What is the probability that the spinner will stop on an even number or a multiple of 3? How do I begin to solve this?
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0answers
18 views

Find Likelihood function of $\theta$ for $f(y|\theta) = \frac{1}{2\theta + 1}$

So, I know the likelihood function is the product of the density from $1$ to $n$. So then it would be $\displaystyle \left(\frac{1}{2\theta + 1}\right)^n$. I just want to see if I am doing that ...
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1answer
15 views

What prior to use given a Poisson likelihood?

I am trying to incorporate a prior into a model I am working on. From available data, I have found that the likelihood follows a Poisson distribution with $\lambda = 1.5$. I have then used R to ...
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1answer
35 views

Unbiased estimator for the sum of numbers

Let $\alpha_1, \dots, \alpha_n \in \mathbb{R}$. We want to approximate the sum as follows $$ S = \sum_{i=1}^{n} \alpha_i \approx \dfrac{n}{c} \sum_{i=1}^{c} \alpha_i, $$ where $\alpha_i$ is picked ...