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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Mean of samples from Exponential distribution

I please ask someone to check if my calculations are right. I have $X_1, ..., X_n$ from a $\mathcal{E}(\lambda): f(x, \lambda) = \lambda e^{-\lambda x}$, $\mathcal n = 50, {\lambda} = 3$. I have to ...
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I need help finding the linear equation for an exponential regression

I've done nearly everything but at the end when I need to $\alpha$ it doesn't work and I'm not sure what's wrong. This is the function: $y = \alpha\beta^x$ I transformed it into a linear equation ...
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18 views

Sufficient statistic for Uniform distribution.

We are given that $ X_i \sim U( 0 , \theta )$ where $\theta$ is unknown. We need to find a sufficient statistic. First we write the conditional distribution : $ f( x_1 ,........ ,x_n | \theta ) = \...
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34 views

How to combine correlated estimates to test variable is > 0?

Let X1 and X2 be two unbiased but correlated Gaussian estimators of a true value x. 1. What is the proper way to combine two observations of X1 and X2 to test whether x > 0? 2. How does the answer ...
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63 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
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99 views

showing SSE of simple regression model is larger than or equal to SSE of multiple regression model

Lets say we have 2 linear regression models: $y_i = B_0 + B_1x_{i1} + \epsilon_i,$ where $\epsilon_i$ follows $N(0,σ_1^2)$ $y_i = B_0 + B_1x_{i1} + B_2x_{i2} + \lambda_i,$ where $ \lambda_i$ follows ...
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12 views

goodness of fit doubt where the next data value is twice

I have a list of data, and the following problem: A particular mobile phone company , determined that by launching its totally innovative new product and never before seen in the market, each day ...
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29 views

CLT for continuous functions of random variables

Let $(X_i)$ be a collection of zero mean, unit variance, real valued random variables (I do not assume that they are iid). Let $\mathcal H$ be a separable RKHS with a bounded kernel $k(x,y)$. Suppose ...
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11 views

ANOVA standard error for treatment difference

can anyone please help me with question b) (I link the image below). It says $\sigma_\Delta=2.241528$ But I got $\sigma_\Delta=2.00488$ I used the equation $\sqrt {\frac {\sigma_i^2}{n_i}+\frac {\...
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25 views

Linear Regression And Coding

I'm revising for my A-level Statistics exam and need to figure out where / when coding has an effect. What I'm currently stuck on is the effect coding has on values for $Sxy$ where $x= y \;$or$\;x $. ...
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566 views

Getting the upper and lower quartiles in data with an even number of observations, or where the quartile lands on a decimal number

I want to draw a box plot, which requires that I know the median, the lower and upper quartiles, and the minimum and maximum values of my data. I understand that the quartiles are simply the value on ...
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404 views

Calculating mode in a grouped frequency distribution

How can I calculate the mode in a grouped frequency distribution when the largest frequency occurs in two or more classes?
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476 views

Discrete Bivariate Distributions, find constant $c$

I am given $f(x,y) = c(x + y)$, and I have to find constant $c$ such that $f(x,y)$ satisfies the conditions of being a joint pmf for two discrete random variables $X$ and $Y$. $x = 1$, 2, 3, and $y =...
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12 views

Meaning of inverse mills ratio

Can someone, please, explain in layman terms what is the meaning of using inverse mills ratio for incidental truncation? I am non-mathematician self-studying this topic and have problems explaining to ...
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417 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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41 views

Picking cards sequentially vs consecutively

We have a pack of 6 cards over the table. Cards are: {A, A}, {B, B}, {C}, {D}. There are 3 players (Papa, Pepe, Popo) sat around the table. Cards are all upside down, so the players cannot see which ...
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27 views

A computer test of a very fast primarily test

Fermat's little theorem states that if $p$ is a prime and $a$ is any integer not divisible by $p$, then $a^{p-1} - 1$ is divisible by $p$. $$a^{p-1}\equiv 1\pmod p$$ This can be used to test if a ...
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28 views

What do attendance figures tell me about regularity? What does the average tell me about individual attendance?

Suppose I have a group of $N$ people, attending a series of $M$ events, and (for simplicity) let's assume the overall attendance happens to be the same at each event, say $A$ people (ranging between 1 ...
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490 views

expectation calculation in probability and statistics

2 four-sided dice are rolled. X = number of odd dice Y = number of even dice Z = number of dice showing 1 or 2 So each of X, Y, Z only takes on the values 0, 1, 2. (a) joint p.m.f. of (X,Y)? joint p....
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31 views

Logistic regression MLE example. What is this “logistic function”?

This is a basic question (I think). I am trying to grasp the idea behind this example, where we define a "logistic function" and use that to work towards the maximum likelihood estimate (MLE). We ...
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how to Obtain standard error of estimates Sy.x and Sx.y. given summated values. please show steps

The data for 10 years on profit earned (Y) by a company and dividend paid (X) by it yielded the following summated values in Kwacha: ΣX =15, ΣY =110, ΣXY = 400, ΣX2 = 250, and ΣY2 = 3200. Obtain i) ...
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21 views

Estimation of suitable significance level for high numbers of experiments

I am a scientist and work with high numbers of experiments. For example: I have tested $ 1000 $ different parameters between $2$ groups. I found $10 $ parameters, which are significantly different ...
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32 views

Understanding a Substep of the Proof for the Law of Total Variance

In the proof for the Law of Total Variance, the following lemma seems to be appealed to (when going from the 2nd to the 3rd step of the proof): $$ E[E[Y^2 \mid X]] = E[\text{Var}[Y \mid X] + [E[Y \...
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System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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36 views

Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
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24 views

Significance of Convex Sets for I-Projection

I have been reviewing the literature on information theoretic methods in statistics, and in particular, the method of I-projections. Given a discrete, finite alphabet $\mathcal{X}$, let $\prod$ denote ...
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91 views

How to find the expected value of this game?

How can I create a function that figures out the expected value for a value that can change. Example: I can flip a (fair) coin $n$ number of times. The pot starts at $\$1$. If I lose, $50$ cents is ...
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34 views

let f(x)=(2x)/9 where 0<x<3 and Find p(x=2)? [closed]

$f(x)=\frac{2x}{9}$ where $0<x<3$ $F(x)=\frac{x^2}{9}$ where $0<x<3$ Find $p(x=2)$? Please help me how can I find the probability when it equal?
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A few questions about a Probability problem involving airline flights and using the phrase “or fewer”.

The question goes as follows: An airline reports that $77.5\%$ of its flights arrive on time. A check of $200$ randomly selected airline flights shows that $140$ of them arrived on time. Estimate ...
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Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and 9....
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13 views

Proof of Random Treatment's Effect on the Causal Regression Function

Consider $C(x)$ to be the outcome a subject would have if they recieve dose $x$ on some real interval. the observed response is given by the random variable relationship $Y = C(X)$. We treat ...
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VC Dimension of star-shaped sets on $\mathbb{R}^2$

Consider a set $U \subset \mathbb{R}^2$ to be star-shaped with respect to a point $\epsilon \in \mathbb{R}^2$ if for every $x \in U$, the segment joining $\epsilon$ and $x$ is contained in $U$. ...
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Statistics. How to find the median of two other medians?

I got median of 226 and another median of 304. Now the question is how to find the overall median? Averaging them is not the correct answer.
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15 views

Determining sample size knowing change effect

I want to figure out the sample size to consider the experiment statistically significant. I did some research and found the following formula: Now, I have $\alpha$, $\beta$, $p_1$, $\Delta$, $k$ ...
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2answers
28 views

Calculate probability from a Negative Binomial Distribution with multiple success scenarios - 7th child born is either 3rd or 4th boy

So the problem I am trying to solve states that, if a newborn is equally likely to be a boy or girl, what is the probability that 1) the third child is the first boy 2) the eighth child is the second ...
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161 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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37 views

Finding the distribution of a minimum of a set of variables

I have been working on a problem where $X_1,...,X_n \sim \mathrm{Exp}(\beta)$ are i.i.d. and $Y = \min\left\{X_1^{0},X_2^{X_1},X_3^{X_1+X_2},...,X_n^{\sum_{i=1}^{n-1} X_i}\right\}$ . I want to find ...
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16 views

Maximum Likelihood Estimate for a uniparametric family

Say we have a simple random sample of size $n$ distributed according to a function of the family: $$ f(x|\theta)=\frac{\theta}{(1+x)^{1+\theta}} $$ where $\theta >0$. We want to compute the ...
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32 views

What is $cov(Y_1,Y_2)$ if $Y_1 = \prod_{i=1}^{n}X_i$,$Y_2 = \prod_{i=2}^{n+1}X_i$ , and $X_{1..n+1}$ are $n+1$ bernoulli-distributed random variables?

$X_1,X_2,...,X_{n+1}$ are $n+1$ random variables that follow a Bernoulli distribution with parameter $p$. We define $Y_1 = \prod_{i=1}^{n}X_i$ and $Y_2 = \prod_{i=2}^{n+1}X_i$. How can I find out ...
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31 views

Probability that n-digit number is divisible by some number(s)?

I have came across a number of problems in our probability course that deal with this kind of question. And for two digit numbers I have always "brute-forced" the solution by writing them all out and ...
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35 views

What is the probability that $X \geq k\cdot Y$, if X and Y are independent and follow the exponential distribution?

$X$ and $Y$ are independent variables that follow the exponential distribution with the same parameter $\beta$. What is the probability $$P(X \geq k\cdot Y)$$ if $k>0$ ? I don't really know where ...
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30 views

Central Limit Theorem: Show that $\mathrm{Pois}(n)$ distribution if approximately Normal if $n$ is a large positive integer

Here is what the solution in the textbook says: Let $S_n=X_1+...+X_n$, with $X_1,X_2,...$ i.i.d. $\sim \mathrm{Pois}(1)$. Then $S_n \sim\mathrm{Pois}(n)$ and for $n$ large, $S_n$ is approximately $...
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Chi square goodness-of-fit test for Uniform distribution using Matlab

I need to test random numbers generators in 1, 2 and 3 dimensions. Criteria of test is that generated numbers are from uniform distribution. I am doing this using Matlab. I'm told to use Chi-2 test ...
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Moments based on a Bootstrapped Mean

Take $X_1,\ldots,X_n$ to be distinct observations, and let $X_1^*,\ldots,X_n^*$ be a bootstrapped sample (given a uniform resampling of cases with replacement) and let $\bar{X_n}^{*} = \frac{1}{n} \...
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Bias of a Particular Estimator

Take $X_1,\ldots,X_n \sim \mathrm{Gamma}(0,\theta)$ And take $\hat{\theta} = \max\{X_1,\ldots,X_n\}$. I want to try to find the bias of this estimator. I can see that $\operatorname{bias}(\hat{\theta})...
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23 views

Trying to find the MLE of $\tau$

Let $\tau = \int x \,dF(x),$ and I want to find the MLE of $\tau$ given $X_1,\ldots,X_n \sim \mathrm{Uniform}(a,b).$ I am not entirely sure, but I would imagine that $\tau = \int x \, dF(x) = \int_a^b ...
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28 views

Minimize the sum of distances between a sample and two “centers”

Suppose we have a set of readings $\{X_{i}\}$, each of which is a real number. What I want is to find 2 numbers, $a$ and $b$, such that minimize the sum of distances between each $X_{i}$ and ...
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22 views

Variance of exponential moving average

I'm not quite sure what the form for the rolling recursive variance should look like. I have that the recursive average is $m_t = \alpha x_t +(1-a)m_{t-1}$. Then would the rolling recursive variance ...
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Using coefficient of variation vs standard deviation to understand trial-to-trial variability

The basic model I have is as follows: I'm studying the response time of a neuron that's affected by two sources of noise: intrinsic noise and synaptic noise (i.e. random fluctuations in the voltage ...