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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Mean and covariance of this stochastic process

We have a process $$ x(t) = \text{atan} (cB(t) + \tan (x(0))), $$ where $c$ is a positive constant, $B(t)$ is a standard Brownian motion, and initial value $x(0)=1$. How to analytically calculate ...
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20 views

Expected values of expected values when using sample statistics

Econometrics class was teaching properties of expected values and showing how they were used in some proofs. I decided to try and proof the variance formula for the variable W [e.g. W=aX+bY] and ran ...
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24 views

Good test statistic

If we have a hypothesis $H_0$ and alternative hypothesis $H_1$, a test statistic $T$ and a data set $x_1, ..., x_n$ taken from some random sample $X_1, ..., X_n$, we use $T(x_1, ..., x_n) = t$ to ...
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24 views

standard deviation for proportion with huge denominator

My question involves finding the standard deviation of a ratio when the denominator is huge. The standard formula $\sqrt{(p*(1-p)/n)}$ yields a number that is almost zero. The context is computing a ...
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0answers
10 views

What is the explanation of Average Variance Extracted in expression f(x) = Ave(y(i)|x(i)∈Nk(x))?

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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24 views

Need Help for Solutions Manual

I am trying to solve all the exercise problems in the book "understanding machine learning: from theory to practice". I am doing so because there is no available solution manual, and the book is such ...
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0answers
28 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
32 views

Is the mean or median a better measure of central tendency?

I was given this question by my friend who said he got it from his teacher (which meant he couldn't clarify the terms in this question). The question is: What is a better measure of central tendency? ...
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12 views

Reduction in mean associated with p-value

I was reading a paper and they reported a mean reduction in parameter $1$ of $5.2\% (p=0.07)$ and a mean reduction in parameter $2$ of $10.7\% (p=0.12)$. Can this be correct? I thought that the ...
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1answer
766 views

Distribution function technique and exponential density

I'm having quite a bit of difficulty with the distribution function technique. If $X_1$ and $X_2$ are independent random variables having exponential densities with parameter $\theta_1$ and $\theta_2$...
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Having trouble with given task [on hold]

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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2answers
1k views

Sample Space & Combinations/Permutations

This is the question: "List the sample space for three individuals chosen at random to vote either Democrat or Republican. List the distinct combinations and permutations." These are what I got for ...
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1answer
28 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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20 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
11 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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1answer
20 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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1answer
55 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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1answer
929 views

If $Y=X\beta+\epsilon$, prove that the least square estimator $\hat\beta$ is independent of $Y-X\hat{\beta}$

Let $Y=X\beta+\epsilon$, where $Y$ is an $n$ by $1$ vector, $X$ is an $n$ by $p$ matrix with full rank and $\epsilon$ is an $n$ by 1 vector of random errors independently and normally distribution ...
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2answers
32 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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0answers
27 views

Averaging with respect to predictive distribution for predicting positive and real-valued ratings

I have given a rating matrix $R$. In this setting, ratings are positive and real-valued scores. $R_{ui}$ is the rating score from user u to item i. My task is to predict a missing rating score $R_{...
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1answer
22 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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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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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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1answer
14 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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1answer
1k views

How to compute SSR with just residuals and Xi?

How do we calculate SSR? I know SSE is the square of residuals all added together, but SSR is a subtraction between prediction for each observation and the population mean. Not sure how calculate SSR. ...
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2answers
38 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
17 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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1answer
256 views

Point of maximal error in the normal approximation of the binomial distribution

I am sorry for the long question! Thanks for taking the time reading the question and for your answers! Context: Let $B_n\sim\text{Binomial(n,p)}$ be the number of successes in $n$ Bernoulli trials ...
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1answer
20 views

Proving almost sure convergence of linear regression coefficients

In the context of simple linear regression, suppose that $\epsilon_i, \ i=1,...,n$ are i.i.d and $ |n^{-1}\sum_{i=1}^{n}x_{i}| \rightarrow |\mu| < \infty$ where n $\rightarrow \infty$ and var(x) = $...
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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 ...
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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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2answers
44 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
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 ...
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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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3answers
966 views

Normal Approximation Binomial

The probability that an appliance is in repair is 0.5. If an apartment complex has 100 such appliances, what is the probability that at least 60 will be in repair? Use the normal approximation to ...
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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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2answers
874 views

Probability at least one will be profitable

There are two investment portfolios, A and B. Portfolio A has 50% probability to be profitable, while Portfolio B has 80% probability to be profitable. The two portfolios are independent. What is the ...
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0answers
25 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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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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0answers
21 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 ...
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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 ...
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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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2answers
846 views

Conditional Probability in Geometric Distribution

Ms. A is taking driving test to get her driving license. The probability of passing a test is 0.3 which remains same no matter how many times she takes the test. Let $X$ is a random variable which ...
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Adjusting normally distributed values for another variable

I asked this on CrossValidated initially but did not get a response so I'll try my chance here. The code parts are not that important and you will understand what I'm asking. Suppose I have this data ...
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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
898 views

Is a 99% upper confidence bound the upper limit of a 99% confidence interval?

I have to find a "99% confidence bound" for a standard deviation. This is not hard. The only question I have is whether this is finding the $\chi^2_{.99}$ value or just the upper bound for the 99% ...
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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-...
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1answer
1k views

Kullback-Leibler divergence of binomial distributions

Suppose $P \sim \mathrm{Bin}(n,p)$ and $Q \sim \mathrm{Bin}(n,q)$. Their Kullback-Leibler divergence is defined by $$D_{KL}(P||Q)=\mathbb{E}_{P}\left[\log\left(\frac{p(x)}{q(x)}\right)\right],$$ with $...
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1answer
28 views

Text book recommendations for statistical estimation theory -specifically MLEs and confidence intervals

I am looking for a textbook on statistical estimation theory. In particular I am interested in a book that explains MLEs and confidence intervals. Preferably accompanied by exercises. The book should ...
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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 ...