Questions tagged [statistics]

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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Having trouble with given task

The random size is distributed according to the normal law, which is an average of 15. Calculate this random size dispersion if it is known that the probability of gaining values ​​from the range [15; ...
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0answers
7 views

What is the explanation of Average Variance Extracted in statistical learning expression given below?

Wikipedia defines it as: average variance extracted (AVE) is a measure of the amount of variance that is captured by a construct in relation to the amount of variance due to measurement error. At my ...
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0answers
18 views

Results for Showing Variables are Uncorrelated

I keep running into implicit assumptions re claims that continuous random variables are uncorrelated. Specifically, I am reading that points on the circle, i.e., the random variables $X, Y= \pm \...
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0answers
6 views

Error propagation to a parameter due to uncertainty in other parameters

I have a function $y = f(x,a,b,c)$ where $a$,$b$,$c$ are parameters. I'm given this curve and I'm trying to extract the value of $a$ given I know $b$ and $c$. However, I have some uncertainty in $b$ ...
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0answers
11 views

Transformation of Wishart distribution

I am a bit confused about transformations of matrix distributions. Say, I have a $n \times n$ matrix $\Sigma$ which has a Wishart distribution $W_n(\Phi^{-1}, n+1)$, so that its p.d.f. is ...
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1answer
25 views

How to combine a number frequency data set into tradional lottery number calculation?

I'm trying to figure out how to combine historical lottery data with a traditional lottery number probability calculation technique to result in a more "accurate" probability of a number being drawn, ...
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1answer
20 views

What does the (absolute) value of the Earth Mover Distance between two histograms mean?

When using the Earth Mover Distance as a metric between two 2D histograms which represent normalized probability distributions, what does the value from the resulting optimization actually mean ? ...
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0answers
6 views

Estimating the mode of a distribution with more than a modal class

$1.)$ How do I estimate by formula and graph histogram, the mode of a grouped frequency distribution in which there are more than one modal class? $2.)$ If after drawing the histogram the modals ...
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0answers
16 views

Sampling from joint distribution by writing its density as a product of conditional densities

In Gelman et al. "Bayesian Data Analysis Ed3" the authors often do the following (e.g. on pg. 65): Given two parameters $\mu$ and $\sigma^2$ and data $y$ joint posterior density $p(\mu, \sigma^2)$ is ...
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2answers
30 views

What's the value of $\lim_{h\to0} P(a\leq X \leq a+h) $?

The book that I am currently reading states that $$ \lim_{h\to0} P(a\leq X \leq a+h) = F(a+0)-F(a)=0 $$ I can't wrap my head around it. According to my understanding, shouldn't it be that $P(a\leq X \...
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1answer
47 views

Conditional Expectation with two random variables

I have two normally distributed independent variables $A\sim N(m,v)$ and $B\sim N(n,w)$. I would like to compute the following conditional expectation: $E[A | c*A+B\geq 0].$ I cannot find an ...
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0answers
15 views

How to find the trimmed mean of an odd amount of numbers?

For a 25% trimmed mean of 50 numbers that means we have to eliminate 12.5 numbers from both the lowest rank and highest rank of the set of 50 numbers. Should I round to 13 or 12 and then do the ...
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38 views

$m$ Men and $n$ Women in a Circle

below are some questions that I would like to understand how to solve efficiently. Thanks in advance! Given $m$ men and $n$ women sitting in a circle: What is the expected number of people sitting ...
2
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2answers
35 views

Efficiency of $\hat{\theta}_{MLE}$ from $\operatorname{Beta}(\theta,1)$

I am working on a problem which asks me to discuss the efficiency of the MLE $\hat{\theta}$ given that $X_1,\ldots,X_n \sim_{iid} \operatorname{Beta}(\theta,1) $. I was able to deduce that $$\hat{\...
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0answers
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CST Exam Question: finding 50th percentile of data [on hold]

What is the median number of hours for a flight? Data given: $25$% of flights are $1$ hour $15$% of flights are $2$ hours $25$% of flights are $3$ hours $35$% of flights are $4$ hours I am ...
2
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3answers
38 views

A level conditional probability

In an A Level maths textbook, the answer to an exercise question involves the use of $$P(B|A')=1-P(B|A)$$ where A and B are two events. However, this equality has never been mentioned elsewhere in ...
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19 views

Different probabilities in an exercise

I've got this following problem solved. A monkey is given 12 blocks: 3 shaped like squares, 3 like rectangles, 3 like triangles, and 3 like circles. If it draws three of each kind in order- say, 3 ...
2
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2answers
39 views

Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
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0answers
35 views

Is my approach to my internship project good? Optimal allocation of product across stores, constrained optimization

(I am new to stack exchange, so I apologize if this is not the right place to ask this. Would love to be pointed to the right direction on where to ask this) Context: I am a CS student currently in a ...
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1answer
19 views

Simulated & Theoretical Values [on hold]

How do I justify the difference in the values obtained via theoretical calculations and simulated calculations with random numbers? Is it always true that theoretical calculations yield higher ...
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0answers
24 views

BLUP is sum of LUP and contrast

In a paper I came across the line "we use the fact that the BLUP of $Z_j$ can be expressed as a sum of any given linear unbiased predictor of $Z_j$ plus a linear unbiased predictor of $0$ i.e a ...
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0answers
20 views

How to statistically compare the outcome of two classification algorithms?

I have two deep learning classifiers and I want to make a test statistic to compare their performance against each other. The two algorithms have been tested on the same 60 samples and have an ...
0
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1answer
25 views

Probability of drawing a spade and then a heart from a deck of cards

What is the probability of drawing a heart and then a spade in 2 successive draws from a standard deck of cards? Do we consider these as independent events thus yielding: $$\Pr(\text{Spade and Heart})...
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1answer
19 views

Mean-squared prediction error why these two different representations are the same?

One representation is E[(y-g(x))^2] and other is $(1/2)m\sum_{i=1}^m (g(x^i)-y^(i))^2$ where m is number of training examples in a training set. First one is from statistical learning book, second is ...
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0answers
22 views

distribution estimation small amount of observations [on hold]

I would like to ask. If I estimate a probability distribution of two or three parameter with only a few (let's say 30 observations) what does it mean for the quality of the estimation? Thank you for ...
0
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2answers
38 views

How to find P(X>Y) for a discrete joint PDF?

I have to find P(X>Y) for the below Joint PDF. $$ P(X=x,Y=y) = \frac{x+y}{12},\space x=1,2;\space\space\space y=1,2; $$ I tried as follows, $$ P(X>Y)= \frac{2+1}{12}=3/12=1/4 $$ $$P(X+Y\le3) = \...
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1answer
12 views

grouped discrete data v.s. grouped continuous data

How to distinguish grouped discrete data and grouped continuous data? I have googled two terms. It seems to me that they are refer to the same thing. I will be appreciated if someone tells me the ...
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0answers
14 views

Numerically stable transformation of logistic mixture to unimodal logistic distribution

I have an $n$-dimensional random vector $\pmb{v}$, where each element is distributed according to some 10-component logistic mixture-distribution. I want to transform each component of the vector into ...
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0answers
41 views

Convergence and asymptotic distribution

Here is the problem I've been working on for a while now: Let $X_1,..., X_n$ be a random sample of size n from a binomial distribution with pdf $f(x|p)=C^m_xp^x(1-p)^{m-x}$, where $m \in Z^+$ and $p\...
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0answers
32 views

An important proposition of the Kolmogorov–Smirnov test

How do I prove the following theorem: The distribution of the Kolmogorov-Smirnov test statistic $D_n$ is, under the Null hypothesis $H_0: F = F_0$, equal for all continuous distribution $F_0$. ...
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1answer
28 views

Loss of uniqueness of quantiles

We know that, if $X$ is a continuous random variable with a strictly increasing distribution function or DF $F(x)$ then, its $p$th quantile is unique. But if the distribution function is non-...
0
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1answer
17 views

Variance of the dot product of a random binary vector and a vector of constants

Let $ q \in \mathbb{R}^N$ and Z a random binary vector s.t. the sum of its elements is n. The variance of the statistic $Z\cdot q = \sum^N _{i=1} Z_i q_i$ should be $ \frac{N-n}{N-1} \times n \times \...
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1answer
15 views

Expected Number of Events over Multiple Poisson Intervals

Let's say I find that in an interval of 5 ns, with event rate 50000/s, that the probability of an event occurring twice in that interval is $$P_2 = {e^{-\lambda} \lambda^{2} \over 2!} = 3.124 \cdot 10^...
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1answer
23 views

Prove that $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not an efficient estimator.

I am asked to show that the unbiased estimator $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not efficient. So far I was able to show that the Rao-Cramer Lower Bound is $\frac{2\sigma^...
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4answers
48 views

Is sample variance always less than or equal to population variance

I was reading this wikipedia article on Bessel's correction: https://en.wikipedia.org/wiki/Bessel%27s_correction. The article says that sample variance is always less than or equal to population ...
1
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1answer
19 views

Cross-Correlation between Variables?

If Pearson's correlation coefficient for variables $X_1, X_2$ is $p_1$, and the correlation coefficient for variables $X_2, X_3$ is $p_2$, can we use this information to reasonably estimate the ...
0
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1answer
19 views

Make two data sets have the same standard deviation

I have two different data sets A and B (37 elements each and they have the same mean), and I would like to modify B to have the same standard deviation as A. I figured to make both have the same ...
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1answer
36 views

Maximum Likelihood Fit of Matrix: How can I find the Covariance Matrix?

I have a 2d Data Matrix $X_{i,j}$ with a corresponding error matrix $E_{i,j}$. I now have a model $X_{i,j} = a_i + b_j + u_{i,j}$ with $u_{i,j}$ being distributed as $N(0, E_{i,j}^2 + \delta_j^2)$. I ...
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0answers
18 views

What does averaging over time means in literature?

Disclaimer: This question is cross posted in here because the answer might be field dependent. Let $h: [0, L]\times \mathbb{Z}^1 \to \mathbb{Z}^1$ be a function, called the height function, and lets ...
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0answers
15 views

Can we use analysis of variance to replace of t test?

I know that we will get the same results whether use analysis of variance(ANOVA) or t tests when we are comparing the mean between two groups. And this two methods have same assumption: normality and ...
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0answers
33 views

Is there any way to make a random sample from the same distribution, i.i.d when it is positively correlated? [on hold]

Is there any way to make a random sample from the same distribution i.i.d when one knows, is given that the subjects have been self selected and and as such are positevely correlated but knows linear ...
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1answer
20 views

Ratio and 90% Confidence interval [on hold]

I have 12 values for T and 12 values for R. I need to find the point estimate (average) of the ratio T/R and 90% confidence interval for the ratio of T/R. I have access to SAS and JMP if its easier to ...
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1answer
25 views

Indicator of two exponential random variables

I am working on the following problem from a statistics qualifying exam. I have attempted progress at the 3 parts but do not feel certain, or satisfied, with my work. I am not sure if $\textit{I}$ am ...
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1answer
48 views

Drawing balls from an urn

I'm preparing for my midterm and I'm having trouble connecting all the dots in the following problem: From an urn containing 2 yellow balls, 3 blue balls and 22 red balls, one ball is drawn, ...
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0answers
36 views

How to approach integration of function involving Probit and Std Norm CDF

I am trying to solve the following definite integral involving some variable $x$ with a domain of $\mathbb R$. $$\int_a^b \Phi^{-1} \big( {\Phi(x) + \Phi(2x) \over 2} \big) dx$$ $$ \text{where } a &...
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1answer
31 views

Not able to solve it for n>2 observations

The relation between $\textrm{AM, GM , HM}$ is given as follows $AM \times HM=( GM )^2$ Does it also follow if the number of observations are more than $2$?
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0answers
41 views

UMP test for testing $f=f_0$ vs $f=f_1$ where $f$ is discrete

As I was studying for my exam I came across the following problem: Let $X$ be a discrete random variable with probability mass function (pmf) given by $f(x) = P(X = x)$. Further, consider the ...
2
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1answer
29 views

Median of 2 sets and median of the sum of the 2 sets

Suppose I have N individuals (N is odd for convenience). Associated with these individuals, two sets (S1 and S2) which contain integer numbers. E.g. S1 is the salary/hour and S2 is the bonus/hour: ...
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1answer
33 views

MLE and UMVUE for $P(X_1=2)$ when $X_i\sim \text{Poisson}(\lambda)$

I've been studying for my exam and came across the following problem: Suppose that $X_1,\ldots,X_n$ is a random sample from a Poisson distribution with mean $\lambda$.(a) Find the maximum ...
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0answers
9 views

Choose a random variable from discrete/continuous distribution. [closed]

Choose a random variable from discrete/continuous distribution. State the parameters and the probability distribution of the random variable. Solve this by using R studio.