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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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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2answers
15 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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4answers
5k 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
38 views

A silly question regarding a badly written exercise for probability equations.

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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21 views

Maximizing Likelihood Function.

Let, $$\mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i,$$ where $\mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf ...
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14 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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15 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. ...
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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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32 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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1answer
65 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...
3
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1answer
638 views

Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
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16 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 ...
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13 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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0answers
23 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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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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30 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
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 & ...
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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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48 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
7 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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1answer
26 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 ...
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1answer
61 views

According to Chebyshev's rule, how many observations should lie within one and a half standard deviations of the mean?

Using the formula : $p = 1 - k^{-2}$ I calculated that $p = 1 - 1.5^{-2} = 0.56$ , which equals to $56\%$. Because I have $24$ data points I go ahead and solve the number of points is $56\%$ of ...
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37 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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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
18 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
32 views

NORMALLY AND INDEPENDENTLY DISTRIBUTED [on hold]

Can someone explain the meaning of NID (normally and independently distributed)?
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1answer
563 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
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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? ...
2
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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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1answer
343 views

Skewness of a difference of random variables?

In this article( http://www.diva-portal.org/smash/get/diva2:302313/FULLTEXT01.pdf )page 28 explains how to derive the skewness of a sum of random variables; I haven't been able to derive this ...
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0answers
75 views

Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z \text{ is Normal(0,1)}$, and $k$ ...
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1answer
19 views

Summation of a fraction containing a summation operator

I came across a proof which had the following sequence: $$\sum_{i=1}^n k_i y_i = \sum_{i=1}^n \frac{(x_i - \bar x)y_i}{\sum_{j=1}^n (x_j - \bar x)^2}$$ where $$k_i = \frac{(x_i - \bar ...
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2answers
5k views

A set having the same mean, median, mode, and range

Is it possible to have a set with the same mean, median, mode, and range? If not, how can the following question be solved: Set $H$ contains five positive integers such that the mean, median, ...
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0answers
11 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
27 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
38 views

Probability function (p.f) of a random variable

If we have a Bernoulli random variable $W$ that is derived from a Variable $T$ (Poisson $\lambda$), by the following rules $W =$ (if $T=0$ then $W=1$ and if $T>0$ then $W=0$), I am having trouble ...
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1answer
33 views

Confidence Interval for a Mean

$15$ credit cards were randomly sampled and analyzed with the following results: (Sample) Mean $= 50.50$ Sample Variance $= 400$ $\sqrt{400} = 20 =$ Standard deviation right? $95\%$ confidence ...
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1answer
201 views

Solving a two-sample z-test for two population proportions is to be performed using the P-value approach.

The null hypothesis is $H_0:P_1=P_2$ and the alternative is $H_a:P_1\neq P_2$. Use the given sample data to find the P-value for the hypothesis test. Give an interpretation of the p-value. $n_1=200$ ...
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0answers
9 views

Finding Y for a Data Set with Std Deviation and Z Score

While only 15% of babies have learned to walk by the age of 8 months, 90% are walking by 13 months of age. If the age at which babies develop the ability to walk can be described by a normal model, ...
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1answer
217 views

Gamblers ruin formula

Hello , I have been reading about gamblers ruin and I found this formula can anyone confirm its accuracy ? I assume they only bet one chip a time
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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
26 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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2answers
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Why the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data?

What makes the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data? For example, if the approximate standard ...
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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
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 ...
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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 ...
0
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1answer
654 views

variance validation

The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean $μ = 74$ and a variance $σ^2 = 8$. Would you still consider $σ^2 = ...
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1answer
607 views

Two-tailed hypothesis test; Why do we multiply p-value by two?

I understand that in a two-tailed hypothesis test, we must multiply the p-value by two. i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed ...