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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Probability of $t$-test rejecting null when $X$ and $Y$ are Cauchy

I have a homework problem that states: The Wilcoxon test is valid for a broad class of distributions, meaning that the actual type I error is as specified. Note that the $t$-test does not have this ...
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28 views

Revenue significance test

I would like to ak if is it possible to make a significance test for revenue from A/B testing when I have these data: ...
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3answers
60 views

Finding probability of a random point [on hold]

Consider a square with sides of length 1 and the bottom left corner at (0;0). Choose a point P randomly within the square. Show that the probability that P is closer to (0;0) than to (0.5, 0.5) is ...
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1answer
6 views

iid random variables (vectors)

If $(X_{1},Y_{1}), (X_{2}, Y_{2}),...,(X_{n}, Y_{n})$ denote a sequence of iid random variables from $(X,Y)$, can I say that each $X_{i}$ is independent from each $Y_{i}$? Or is it just for the ...
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1answer
14 views

Proving negative natural log of Beta($\alpha$, 1) distribution is an exponential distribution

I'm looking to prove that taking the negative natural logarithm of a Beta distribution with parameters $\alpha$ and $\beta=1$ is an exponential functions. I've found two different proofs, both of ...
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11 views

Use of the chi-square test with low expected value

I have a question about the use of the chi-square test. I've been told that use of the chi-squared test for independence requires that most of the expected values be over 5. What is worrying is ...
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2answers
14 views

Time series analysis. Understanding the arma model [on hold]

Determine wether $Y_t= 0.7 + 0.4Y_{t-1} + 0.12Y_{t-2} +Z_t$ is a stationary process.
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16 views

finding distribution based on 1st and 2nd moment

I am to determine the unbiased probability densities $p_1 (x)$ and $p_2 (x)$ given the only constraints that the magnitude of the first moment of p1 is fixed (i.e. $<x> = a$ for some real a) and ...
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1answer
8 views

how to calculate central value of some data?

how to calculate central value of the following sets: I'm thinking is the same as the median. Is it? {-2, -1, 3, 5, 7 , 1, 3 , 6, 2 , -1, -5} and ...
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8 views

Conceptual doubt between observed values and samples

first of all, I know almost nothing about statistics (and I do no like it or I've never found a good reference). Apparently it is usual to write capital letters to denote random variables and small ...
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1answer
17 views

Probabilities through Simulations

Ten percent of people are left handed. If 15 people are randomly selected, how many would be left handed and how many would not be left handed?
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26 views

How to prove a prediction system is more accurate than random

I've created a system that tries to predict that outcome of golf tournaments, suppose there are 97 competitors for a tournament and I can look back and compare my predictions with the actual ...
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3answers
38 views

Reliability of linear regression to predict future

When we have a set of data, where X is the cause, and Y is the effect, we can use linear regression to predict values for Y, based on values of X. I have learned that you may only safely apply this ...
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1answer
13 views

Mean of data with uncertainties and confidence interval

My question sounds rather simple, but sadly I have not been able to find an answer online (which is rather strange as this seems very basic, so I apologize if I simply haven't looked well enough) The ...
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2answers
35 views

Finding conditional probability [on hold]

Two floppies are selected at random without replacement from a box containing $7$ good and $3$ defective floppies. Let $A$ be the event that the first floppy drawn is defective, and let $B$ be the ...
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1answer
27 views

Finding error variance and confidence interval

Two new types of petrol, called premium and super, are introduced in the market, and their manufacturers claim that they give extra mileage. Following data were obtained on extra mileage which is ...
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17 views

test the mean difference of two Gamma distributions

If I don't have the data, but only have the estimates of mean and variance of two independent Gamma distributions. What type of test I can use to test the null hypothesis $\mu_1=\mu_2$?
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1answer
22 views

Finding confidence interval

A random sample of $700$ units from a large consignment showed that $200$ were damaged.how can we find the $95 \%$ confidence interval for the proportion of damaged unit in the consignment.
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1answer
19 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
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2answers
33 views

question about probability problem

How is the last column calculated? I don't understand, and I don't understand the explanation. $P(A \cap B)$ is calculated by $P(A)P(B\mid A),$ right? How is $P(A\mid B)$ calculated? Thanks
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1answer
18 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
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1answer
28 views

Odds of winning more than 50% of many bet of different %

I made a bet with a friend and I would like to know if I'm ahead or not. We have a package of 6 games. Each game have a different probability of a team to win. If it's a tie 3-3, it's a push. If it's ...
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0answers
7 views

Asymptotic Properties of Transformation of Estimators

I'm trying to find a good explanation/proof for the following statement: If $ \sqrt{n}({\hat{\theta}} - \theta) \to^{d} N(0, \sigma^2)$, then $ \sqrt{n}({g(\hat{\theta}}) - g(\theta)) \to^{d} N(0, ...
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1answer
12 views

(Linear regression and data points) Does anyone have links to good tutorials, or advice? [on hold]

I have gathered several sets of 1500 data points. I am trying to import the data and run a simple linear regression model on the points. I have found very vague tutorials to teach me how to do this. I ...
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1answer
11 views

Integrate a PDF over a set

This is part of a proof of Chebyshev's inequality, but there's one line I'm just trying to clarify my intuition of: $f(x)$ is the pdf and $g(x)$ is a non-negative function. $$ \int\limits_{x: g(x) ...
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16 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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16 views

WHY?The Squared Euclidean Imbalance are equal to Fourier coefficients

I'm reading the classical paper about distinguishing attack, How Far Can We Go Beyond Linear Cryptanalysis ,Thomas Baign`eres, Pascal Junod, and Serge Vaudenay. The only proposition I don't ...
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19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
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0answers
23 views

Which hypothesis test to use

Two identical machines are used to make a special coin. We want to know if they have the same variability. A random sample is taken from each machine : $$ \begin{matrix} MachineA & 135 & ...
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1answer
18 views

Bound for deviation from mean of binomial random variable

Let $S_n$ denote a binomial $(n,p)$ random variable. The problem is to show that there is a constant $C$ depending only on $p$ such that for all $\epsilon > 0,$ $$P(|S_n/n - p| > \epsilon) \le ...
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0answers
26 views

Can I estimate Variance of Gamma from Negative Binomial distributed data, given NB is Poisson-Gamma mixture

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a mixture of Poisson and Gamma. The variance of this Gamma distribution is ...
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2answers
28 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
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1answer
18 views

What to do with the boundary values of a bin in a histogram?

Suppose I want to make a simple frequency histogram of the following data: 3 3 4 5 5 6 7 7 8 10 11 I'm supposed to use bins of size 5, starting with zero. Here's my question: Is there a standard ...
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15 views

Covariance matrix with constant diagonal

Is there a term for covariance matrices with constant diagonal (variance of every entry being equal)?
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0answers
20 views

Confidence Interval [on hold]

The data for this problem is in the image below. I attempted this problem in the work shown above. I understand the formula to find the confidence interval for this data, but I am stuck at a certain ...
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0answers
10 views

Find the order statistics joint distribution $(X_{(1)}, X_{(2)} - X_{(1)}, \cdots, X_{(n)} - X_{(n-1)})$ [on hold]

Let $X_1, \cdots, X_n \sim U(0,1)$ i.i.d. Then, Find the joint distribution of $(X_{(1)}, X_{(2)} - X_{(1)}, \cdots, X_{(n)} - X_{(n-1)})$ Could you give any hint for this problem?
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1answer
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how to test a hypothesis when given population mean and standard deviation is known?

the mean weight of a 6 year old child is 49.3 lbs. another study indicated that in a sample of 196 six year olds the mean weight was 51.5 lbs. Assume a population standard deviation of 14. use the ...
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1answer
31 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
30 views

A circle with radius $r$ has $k$ points within its radius. What is the min number of points a square with side $2r$ need to maintain this?

Consider that I have a circumference $c$ with radius $r$ that has $k$ points within its distance. As shown by the following graphic: For perfomance issues, I need to normalize this circumference ...
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17 views

Variance of a linear combination of correlated variables

$\newcommand{\var}{\operatorname{var}} \newcommand{\cov}{\operatorname{cov}}$ I want to calculate the variance of the following linear combination: $$ \begin{align} & \var\left[X_3 - \frac{X_1 + ...
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1answer
15 views

Variance formula - need verifying [on hold]

$$D^2(X)=D^2(E(X|Y))+E(D^2(X|Y))$$ Can someone verify that this is true and why?
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0answers
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Statistics question?

In a survey in town A, 17 out of 20 people had seen a certain movie. In a survey in town B, 19 out of 30 had seen the movie. Assuming simple random samples, test the hypothesis that the proportion of ...
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0answers
9 views

$\chi^2$ P values comes to be zero

I want to find the P Value of $\chi^2$ (Pearson), in order to see if there is a significant difference between the given two following distributions: ...
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0answers
17 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
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0answers
6 views

Random numbers with groupings

I have 9 items, I need to pick a random item and I would like each item to be just as likely as the others to be chosen. I'm assuming my RNG is perfectly. Currently I am picking a random number ...
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Poisson distribution confidence intervals and hypothesis

I think I have A and B correct but I have troubles with parts C and D. A) What is the p value if we suppose the following : finding golden apples in a tree follows a Poisson P(2) with $\lambda = 2$ ? ...
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Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
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29 views

HELP! Marginal Probability Density Function of X and Y (pictures).

I understand the formulas for finding the marginal PDF of X and Y, however, in this example, how do we get from this to that: Thanks a lot!
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Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
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Doing a confirmative factor analysis for Technology Acceptance Modell

i'm fairly new to statistics an need to check if the TechnologyAcceptanceModel(can be seen here: http://istheory.byu.edu/w/images/9/90/Tam.JPG) fits my data. I extended it by a Factor of Usability. ...