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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Silly question regarding probability?

A probability exercise which I believe that it is written in slightly dodgy manner as I have trouble finding a solution for it. The way I go about the solution is to have the problem split into three ...
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1answer
17 views

Power of hypothesis test

Consider two probability density functions on [0,1]:$f_0(x) = 1$, and $f_1(x) = 2x$. Among all the tests of the null hypothesis $H_0 : X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with a ...
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11 views

Central Limit Theorem and Inhomogeneous Poisson Distributions?

A random variable $X$ is said to have a (homogeneous) Poisson distribution with parameter $\lambda$ if for $k = 0,1,2,\ldots$ the probability distribution function is given by $f(X = k) = ...
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1answer
14 views

Distribution of residual term in regression.

In regression analysis for classical linear regression model the residual term is independent of x and y and normally distributed and it is a random variable but i found somewhere written u~N and ...
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2answers
25 views

How do I specify the inverse of a correlation matrix?

To specify a correlation matrix $\in \mathbf{R}^{n\times n}$. There are $n(n-1)/2$ free elements. If I wanted to specify a matrix that is the inverse of some correlation matrix, how should I specify ...
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0answers
18 views

Calculating statistics of “collisions”

I have n sources of events. Each of them is independent, and they all have the same probability $P$ to produce an event during a time $T$. During the time $T$, they have at most 1 event (meaning that ...
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12 views

calculating $E(\Pi_{i=1}^{n}\displaystyle\frac{X_i}{X_{(n)}})$. [on hold]

suppose $X_1,X_2,...,X_n$ be a random sample of $U(0,\theta)$. how can I calculate $E(\Pi_{i=1}^{n}\displaystyle\frac{X_i}{X_{(n)}})$. $X_{(n)}$ = $max_{1\leq i \leq n}X_i$
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2answers
17 views

Probability: Reading tables and using the data from them?

Alright probability is not as hard as I imagined yet I strugle with reading tables and applying them to the formulas. The question bellow has a table with 3 rows and 3 collumns and I am asked to see ...
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1answer
46 views

A silly question regarding a badly written exercise for probability equations. [on hold]

I am doing some exercises and this silly question is bothering me even though I am familiar with probability theory and Bayes law but this question is written in a rather peculiar manner I have no ...
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0answers
16 views

“Confidence Intervals” with *very* small samples.

I am collecting performance data for an application, I gather results each version. Given that the measurement is imprecise, I'd like to calculate a confidence interval for the performance data. ...
1
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0answers
17 views

Find the value of the power of a hypothesis test

Consider two probability density functions on [0,1]:$f_0(x) = 1$, and $f_1(x) = 2x$. Among all the tests of the null hypothesis $H_0 : X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with a ...
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0answers
7 views

How to get a balanced score between three numbers of differing sizes?

I have four items, each scored in three different categories. I would like to end up with one "final" score, which is a balance between the three scores. ...
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0answers
35 views

How can I use mathematics to determine if my data is accurate?

Imagine I have an application which logs data about interactions in a shop. For example, whenever a customer comes in and purchases an item, the shopkeeper logs what item was purchased and the ...
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0answers
17 views

martingale betting probability

What would be the probability of reaching a total of $40$ at a 50-50game using the martingale system with $20$ and standardbet 0.5? is it just 50% ? or is this strategy better/worse in short term ...
2
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0answers
14 views

statistical tolerance limits

My Question is: What statistical methods can be used to examine whether a selection of tolerance limits of data makes sense, i.e. validation of tolerance limits? My solution: I would analyse how many ...
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2answers
10 views

Does this follow a binomial distribution?

Q. Four roads start from a junction. Only one of them leads to a mall. The remaining roads ultimately lead back to the starting point. A person not familiar with these roads wants to try the different ...
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0answers
34 views

How can I calculate accuracy of an estimate based on a subset of data?

For example, say I'm trying to determine the average number of steps taken in a day for a population of 100M people. I know the exact number of steps for a random 25M subset of the 100M group (2500 ...
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1answer
11 views

Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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0answers
9 views

How to estimate the duration of the path?

Let $G=<V,E>$ $p$ - sequence of vertices and edges For each edge $(u,v)\in E$ there is information about the transition time from $u$ to $v$ represented as a set of values $T=\{t_0, t_1, ... ...
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2answers
11 views

what is the difference between multi-linear coefficient and multiple linear regression

what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression. I am confused at this point.
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0answers
54 views

Finding the right $\sigma$-algebra. Question on uncertainty related to the secretary problem.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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0answers
49 views

Show that $\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ [duplicate]

$\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ Tried: here suppose N is poission distribution with parameter n $\lim_{n\rightarrow \infty} \sum_{i=0}^{n} ...
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0answers
24 views

Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
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0answers
19 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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0answers
9 views

Does marginalizing on a Bayesian network preserve its original independence assumptions?

I know that marginalizing over a Bayesian network causes changes to the graph (e.g. marginalizing node $c$ in the V-structure given by $a \rightarrow c \leftarrow b$ results in $a$ and $b$ being ...
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0answers
35 views

NORMALLY AND INDEPENDENTLY DISTRIBUTED [on hold]

In regression analysis for classical linear regression model the residual term is independent of x and y and normally distributed and it is a random variable but i found somewhere written u~N and ...
3
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0answers
38 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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1answer
29 views

A certain store sells $31$ different flavors of ice cream. How many different $3$-scoop cones are possible if :

A certain store sells $31$ different flavors of ice cream .How many different $3$-scoop cones are possible if : a) each flavor must be different and the order of flavors is unimportant? ...
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1answer
27 views

bivariate density function - components are not independent

I want to show that the random variables of the density function $$\begin{cases}π^{−1} & :\textsf{if }x^2 + y^2 < 1\\0 & :\textsf{elsewhere}\end{cases}$$ ... is dependent. However, I ...
2
votes
2answers
25 views

problem with Combination and Permutation

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated: a)if each couple is to sit together? (8)(1)(6)(1)(4)(1)(2)(1) b)if all men sit together? ...
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0answers
12 views

Algorithm creates representative set of data [on hold]

I have algorithm to analyze and make it easier to implement in programming language (Python). We have table with data and we want to select only representative part. It looks like: ID_PRODUCT | ...
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0answers
29 views

Problem in selecting hypothesis

This question form edexcel S2 past paper with answer copied from the marking scheme. Here why we put alternative hypothesis as $p>0.5$ without using $p<0.5$ because in question it mention as ...
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1answer
43 views

Finding the conditional probability

enter image description here Let $(X,Y)$ be a two-dimensional stochastic vector with density $$ f_{X,Y}(x,y) = \begin{cases} \dfrac{e^{-y}} y & \text{if } 0<x<y, \\[4pt] \,\,\,\, 0 & ...
0
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1answer
28 views

sample standard deviation given population standard deviation

How do you find the sample standard deviation when given the population standard deviation? What formula do you use? If you can make up an example that would be great.
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0answers
27 views

Ratio of two normal random variables with the same mean and same standard deviation

I would like to compute the probability density function of $Z = \dfrac{X}{Y}$ with $X$ and $Y$ following a non-standard normal distribution with the same parameters (same mean and variance).
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0answers
18 views

Sample standard deviation from population standard deviation [on hold]

How do you find the sample standard deviation when given the population standard deviation and no other data? What formula do you use?
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0answers
7 views

Calculate popularity (frequency) of an item sold from multidimensional array [on hold]

I have an array extracted from my database online where i store id status: sold / not sold time: day that the item sold or not price sold and the output example is given below ...
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0answers
32 views

Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
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1answer
18 views

Stochastic Variables

There are $7$ people in a room. $4$ male & $3$ female. $2$ persons are chosen randomly. The probability of "doing something right" if $2$ females are chosen is $0.6\,$. When 2 males are chosen the ...
0
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1answer
28 views

Can the standard deviation be as large as the value itself

While tutoring, a statistics student presented me with this problem which appears to have an unrealistic standard deviation value. "the mean age of women is 21 years old. A random sample of 30 is ...
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0answers
10 views

How do I find the type I error of these solutions

I found part of an exercise where: $\bar{x} = 0.53$, $s=0.056$, $n=6$. $H_0: \mu = 0.5$ $H_1: \mu < 0.5$ I found that $t=1.31559$ and this had to be checked for 90% en 80% certainty. ...
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0answers
20 views

Beginner Q: Degrees of freedom on chi squared tests and estimating parameters

Suppose that I am conducting a goodness of fit test to see whether a dataset follows a binomial distribution and I have 6 observed frequencies. From this, I estimate the value of p, and work out 6 ...
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0answers
20 views

I need to find the conditional probability and show that XY and Y are independent [on hold]

I need help with the following as I don't know how to calculate it. I however did most of it and the formulas are there as well: enter image description here Can anyone please help me finish it?
-1
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0answers
28 views

Problems deriving probability generating function for the negative binomial distribution [on hold]

My problem is the following: Part A.a I can't get the moment generating function to be what it states in the exercise. And I found other people asking the same question but they get a different ...
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0answers
92 views

Mathematics Homework 2 Question 8d :What is the probability you and your partner are now able to meet the new deadline? [on hold]

You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned ...
1
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1answer
22 views

For $X_1,X_2, \cdots,X_n$ ~ Poisson($\lambda$), find UMVU estimator for $\lambda^k$ ($k=1,2$,…)

I have some questions about this problem as I'm reviewing for a qual. Our TA provided us with a solution, but I don't understand what is going on: So it looks like they are trying to find an ...
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0answers
17 views

Finding Asymptotic Confidence Interval with a condition

I am trying to solve the following problem: Let $X_1, X_2$, and $X_3$ be random variables from the following joint pmf: $$f_{X_1,X_2,X_3}(x_1, x_2, x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} ...
11
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3answers
344 views

Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n ...
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0answers
20 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Let $x_1,x_2,...,x_n,x_{n+1}$ be $n+1$ i.i.d. random variable with non-negative ...
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18 views

finding the sufficient and ancillary statistics

I am trying to find the sufficient and Ancillary statistics for the following problem: suppose $(X_1, Y_1). . . ,(X_n, Y_n)$ be iid random vectors from the pdf: $f_{X,Y} (x, y) = {\frac{1}{2π ...