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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Piece-wise probability density and cumulative distribution function exercise

Given a random variable $X$ with the density function: $f(x) = a$ if $0 \leq x \leq b$ and $f(x) = b$ if $b < x < a + b$ I want to solve the following exercises regarding this ...
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0answers
22 views

Rao-Blackwell theorem and conditional distribution

Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based ...
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1answer
23 views

Kurtosis of sum of Independent Random Variables

Suppose that $X$ and $Y$ are independent random variables with different expected values and variances. Suppose we define kurtosis as $$Kurt(X)=\frac{E[(X- \mu)^4]}{E[(X- \mu)^2]^2}$$ My question is ...
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1answer
19 views

Derivation of the moment generating function of the geometric distribution - why is this wrong?

Let $P(k,X)=p(1-p)^{k-1}$. When deriving the moment generating function I start off as follows: $E[e^{kt}X]=\sum\limits_{k=1}^{\infty}e^{kt}p(1-p)^{k-1}$. How I end up rearranging this is as ...
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18 views

Bayes estimates

How do I attempt at solving this problem? Could I use proportionality? Bayes estimate of parameter of lambda with Poisson likelihood with x = (1,5,4,4) and gamma prior for lambda with mean = 2 and ...
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1answer
41 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
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0answers
9 views

Can an average calculated with a neutral weight be different from a simple average?

I'm working with a data set and just found that some averages calculated with a neutral average (due to the sample size being too small) is different from the simple average result. Is this possible?
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29 views

Integration of gaussian divided by square root of -log(1-x) - does the Meijer G function help me?

After some modelling of my data I came to the following integral: $$ \int_0^{1}\dfrac{exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\,\sigma^2}\right)}}{\sqrt{-\log{(1-x)}}} $$ I cannot solve it, and ...
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1answer
52 views

Conditional probability to Conditional expectation

My first related question is this link. enter link description here Expected value of geometric distribution is $$c=E(X)$$ $$c=0\times p+(1+c)\times q =q+cq$$ $$c=\frac{q}{p}$$ when $X= number \ of ...
2
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1answer
22 views

Expected value of geometric distribution

I watched $Statistics 110$ in Harvard University through YouTube. At lecture 9, I understand that the expected value of geometric distribution is $$\sum\limits_{k=0}^{\infty} ...
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1answer
15 views

Comparing Binomial R.V.s to Poisson and Normal Approximations

I'm hoping someone can clarify this problem I was given, because I'm not quite sure if my steps are correct. Q: Compute P(X <= 10) when X is a binomial random variable with parameters n = 100, p = ...
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1answer
11 views

Distribution of exponential(X/c)

Suppose $X \sim Exponential(\lambda)$. That is, the PDF for $X$ is $f_X(x)=\lambda \cdot e^{-\lambda x}$, $x\ge 0$, and the CDF of $X$ is $F_X (x)=\int_{-\infty}^x f_X(x)=1-e^{-\lambda x}$, $x\ge ...
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0answers
11 views

Given a sample determine using Chi-squared test whether these values fit in an EXPONENTIAL distribution

Here I've got such a problem. I was given $n = 20$ values for time of good functioning of a robot between two consecutive defects. 1200, 1432, 1502, 1100, 3286, 4235, 1149, 5236, 2234, ...
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2answers
28 views

In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
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11 views

Question about the relationship between correlations

I am trying to figure out the relationship between correlations of variables where one of the variables defined as the difference between two other variables. I have variables x and z, which are ...
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2answers
58 views

How do correlation and causality effect this scenario?

I came up with a real world problem that I don't need to solve, but was intrigued by nonetheless. Imagine I was trying to figure out if a supermarket chain took credit as well as cash. My wife had ...
2
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1answer
23 views

make the mean of 2 data sets the same

I am in the process of computing my student's final scores and for some reason I couldn't figure this step out: If I have 2 data sets of scores in the range [0,40] but one data set has a different ...
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2answers
21 views

Unbiased Estimator test

I'm in need of help with this problem. I've been reviewing for an exam and I'm terrible with estimations. $X_1,X_2,\dots,X_n$ be an i.i.d sample from continuous uniform distribution on the interval ...
2
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2answers
32 views

Calculate the confidence interval of parameter of exponential distribution?

How can I calculate the confidence interval for parameter $\alpha$ of exponential distribution ? I think I can use test-t. Knowing that: $$mean = {1\over\alpha}$$ I found that : $${1\over ...
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2answers
23 views

Estimating population standard deviation with sample standard deviation

At 4:30 of this video the author decided to estimate the standard deviation of the population with sample standard deviation (sample size was $100$). In the next video, the author mentioned that it ...
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0answers
21 views

Smallest square problem, $A^*A$ singular?

In our numerics class, we have to solve the smallest square problem $Ax = b$ with $$A = \left( \begin{matrix} 1 & 3 &-4\\ 3 & 9 & -2\\ 4 & 12 & -6\\ 2 & 6 & 2 ...
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1answer
20 views

Null Hypothesis question phrasing

If I want to determine if people like tea or coffee more and conduct a simple random survey of about 50 people, I have some confusion about how to phrase the question to the people and how that ...
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0answers
13 views

How to calculate effect of different variables/parameters on a quantity?

I am developing a game for iOS. In the game I have around eight different parameters that directly affect the score of the player. We can say that these eight variables decide the difficulty of the ...
2
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1answer
38 views

How many tosses are necessary that $n$ players produce pairwise different numbers of tosses with the result “heads”?

$n$ players toss a fair coin. The number of tosses with the result "heads" is recognized for each player. The game stops if the numbers are pairwise different. Let $X$ be the number of tosses for ...
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1answer
5 views

Variance function is variance stabilising

Y has mean u and variance function V(u). If $V(u) = \alpha.u^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that Var(h(Y)) is approximately constant. I tried to prove it computing ...
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1answer
17 views

Finding an interval estimate for $\mu$ given a sample size and variance

I'm in a statistics class and am doing a problem for homework about confidence intervals. I don't really know what it's asking though or when I've even reached a valid solution. The problem says: ...
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2answers
47 views

Rolling two dice… [closed]

Let $A_n$ be the number of fives, $B_n$ the number of sixes and $C_n$ the number of eights in $n$ rolls of two dices. For which n do we have: $E(A_n) < E(min(B_n,C_n))$ ?
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1answer
35 views

Confidence Interval for Nonlinear Regression using F-Test - lmfit

I am trying to understand the implementation for the lmfit confidence interval calculation - in the docs it is stated: "The F-test is used to compare our null model, which is the best fit we have ...
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0answers
13 views

Probability of error in estimation

We are given the cumulative partition function of the rainy days in march and july. We have a paper, without the name of the month (but we know it is one of these two), on which 14 rainy days are ...
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1answer
96 views

Approximate th Probability of a Sum of 16 Independent Uniform R.V.s

This question has to do with the Central Limit Theorem, uniform random variables, and cumulative distribution functions, I believe, but I'm not quite sure how to apply them all in the proper way. Q: ...
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22 views

Is the following data set possible?

"Is it possible to create a data set where $\bar{x}=30.0$, range $R=10$, and variance $s^2=40.0$?" I feel sort of dumb asking this question, but I'm not quite sure I'm on the right track. I know that ...
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1answer
24 views

How to calculate the probility of 2 independent events of having same value?

We are learning to calculate the probability of sums and difference of random numbers. Here is the problem: One athlete knows from past experience that the distances of his javelin throws follow a ...
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2answers
25 views

Binormal Probability [closed]

Q. At a certain time, Hawaii contained 770,000 people, 60% Asian, 39% White and 1% Black, if a random sample of 7 persons was drawn. What are the chance that a majority are Asians? Can someone ...
2
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1answer
38 views

Validity of conditional CDF proof via PDF integral

Given the question: $$\text{Show that}\ F_X(x\mid A) = \dfrac{\Pr(A\mid X\leq x)}{\Pr(A)}\cdot F_X(x)$$ I have seen the solution via probabilities 'directly'. My question is whether the following ...
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1answer
24 views

Is the event is plausible or not?

An atlete specialized in long jump events jumps an average of $\bar x=7.91m$ in $12$ trials. The standard error of the mean jump distance in these trials is $0.2m$. Is it plausible that when the ...
0
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1answer
18 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fittet parameters and there after refit ...
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0answers
8 views

Sampling With Replacement No-Repeats

I am about to do a survey on a population of 250 individuals. I will be performing a Two Sample T Test For A Difference Between Two Populations but am having trouble meeting the conditions for the 10% ...
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0answers
15 views

Trying to find the specific name for such a function.

We know that if $\frac{f(x + t)}{f(x)}$ increases in $x$ for $x > 0$ and $t > 0$, then $f$ is said to be a $\log$ convex function. But if we have $\frac{f(xz)}{f(x)}$ increasing in $x$ for $x ...
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2answers
36 views

Probability of being highest OR second highest draw from a distribution

In my situation, I have a distribution F(x) over some compact interval. Say I take $n$ iid draws from the distribution. I want to find the probability that one draw, $x_i$, is the highest of the $n$ ...
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2answers
17 views

Using Poisson Distribution Method

Q) The question is as follows: suppose that a book of 200 pages contains 20 printing mistakes.Assume that are errors are randomly distributed throughout the book and x , the number of errors per page ...
0
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1answer
32 views

MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
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0answers
12 views

Correlated random effects?

Say I have a linear model where two random effects are possibly correlated, do I still have a random effect matrix for each of them? for instance if $Z_1X=U_1,Z_2W=U_2$ are two possibly correlated ...
0
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1answer
23 views

Probability and statistics Hypothesis testing

I'm not sure if this is the right place to publish this question, but I am trying to understand something about hypothesis testing. recently in a lecture on the subject a professor emphasized how ...
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0answers
10 views

Allan Deviation Explanation

I'm not sure if this is the correct stackexchange for this but I'm looking for an explanation of Allan Deviation or Allan Variance. I'm trying to learn it for research I'm working in for analyzing ...
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30 views

Chances in combinatorics [closed]

There are 124 apples, you and 9 more people are taking them. What are the chances for you to take the one that you wanted?
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1answer
8 views

What can we say about the concentration around 0 linear transformation of Gaussian random variables?

I have a matrix $X \in \mathbb{R}^{n \times m}$ such that each $A_{ij}$ is a Gaussian with mean $0$ and variance $1$. We have $m > n$. I also have a vector $v \in \mathbb{R}^m$ such that $||v||_2 ...
0
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2answers
43 views

Quantify preference: better, similar, worse

I am asking a sample of men to compare the perceived sharpness of two blades (A and B). One blade may be only sharper, duller, or similar to the other. How do I make sense of the following examples: ...
0
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2answers
30 views

Probability Normal Disitrubution, P(X<=1.5)

I think this is simple but I'm not sure how it works (I probably just dozed off while reading this part). The important info is that X is a normal random variable with mean 3 and standard deviation ...
0
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1answer
33 views

Is there a simpler calculation for the probability of exactly x occurrences over n trials?

What I've worked out for this, where x is the specific number of occurrences, n is the number of trials and p is the probability of an occurrence during a single trial, is this: \begin{equation}‎ ...
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1answer
9 views

Shift graph towards mean with time?

Say I have a bar graph with one dependent variable and one independent variable. Given a time $t$, I essentially want to modify the given graph so that as $t -> \infty$, all the bars become equal ...