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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Help understanding interaction and effect.

I am reading a research paper in which the authors designed three bots and wanted to determine what bot was most human-like (link is at the bottom). To do this they had subjects play against each bot ...
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2answers
227 views

the profit of pizza

The number of pizzas delivered to university students each month is a random variable with the following probability distribution $x=\{0,1,2,3\}$ and $$P(X=x)=\{.1,.3,.4,.2\}$$ respectively. If the ...
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56 views

Is Zn a Martingale with mean 1?

Consider a sequence of independent tosses of a coin, and let $P_h$ be the probability of a head on any toss. Let $A$ be the hypothesis that $P_h = a$, and let $B$ be the hypothesis that $P_h = b$. Let ...
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42 views

Multivariate convergence in distribution

Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$. Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$ Prove that convergence in Distribution of $$ \sqrt(n) ...
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36 views

Expected Sum of Weights after Drawing Without Replacement

We have an urn containing $k$ balls where for all $i:1\le i\le k$, the ball $b_i$ has the size $s_i$ that determines its probability to be drawn. For instance, a ball $b_i$ with size $s_i=3$ is ...
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1answer
51 views

Find the limiting distribution of the following random variable

Let $X_1,X_2,...$ Be independent random variables with common density: $$f_X(x)=\alpha x^{-(\alpha+1)}. x>1$$ Where $\alpha>0$. Define a new sequence of random variables: ...
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1answer
83 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
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1answer
27 views

CLT approximation

Let $X_1,\ldots,X_{735},Y_1,\ldots,Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$ and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Find $P(\sum_{i=1}^{735} X_i ...
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16 views

wilcoxon sum test

Question A group of researchers believe children to vegetarians weigh more than those who eat meat. They gathered data: Vegetarian children: 3.4, 3.3, 4.2, 5.1, 4.6, 3.0 Meat eaters:: 4.0, 4.6, ...
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2answers
135 views

craps probability question

The following exercise is best solved with a computer. The probability of winning a game of craps (a dice-throwing game played in casinos) is 244/495. a. What is the probability of winning 5 or more ...
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25 views

Prove the relatinship between Beta Distribution and Bionomial Distribution

Prove that $$ \sum_{k=0}^{x}{n\choose k} p^k (a-p)^{n-k} = (n-x){n\choose x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt $$ (Hint : Integrate by parts or differentiate both sides with respect to $p$) From the ...
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107 views

How to increase mean of set of numbers

Lets say: $X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$ If I have to increase mean of set $X$ by $3$, each number in the set has to be increase ...
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1answer
44 views

Show that $E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $ for a martingale with $Z_0=0$

I was just wondering, if we let $(Z_n)_{n\geq 0}$be a martingale with $Z_0=0$, is it true then $$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$ Please let me know and if it is true, can someone show ...
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2answers
90 views

Probabilities and Binomial Distributions

Suppose $X$ is a binomial random variable with $n = 25$ and $p = 0.7$. Use Excel or Minitab to find the following. Please give the answers to five decimal places. $$P(X = 16) = ?$$ I used the excel ...
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25 views

What is the rationale behind ROC curves?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate. 1) I don't understand how for a given statistical learning model, you ...
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28 views

Why is the $p$-value in hypothesis testing written with a supremum?

I noticed that there is a definition of the $p$-value in my textbook is defined says the following: I have no idea why it is written with a supremum. I've spent hours pondering this. Does anyone ...
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1answer
76 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
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1answer
40 views

variance of minimum of squared exponential random variable

Given $Y_1 $ to $Y_n$ are exponential r.v's with mean $\theta$ find $\operatorname{var}[\min(Y^2 )]$ with the help of gamma distribution. attempt: $\min(Y) $ is exponential with $(\text{mean} = ...
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1answer
57 views

Bayesian Probability Question - Parameter Estimation

I would like help on the following question and I will show my work. Here is the question in my notes and I will follow up with my work: Q: Suppose a forest is segmented into strips, referred to as ...
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2answers
157 views

Probability of always rolling 6 on a dice

Suppose I roll a six-sided die $10$ times and each time it shows a $6$. What is the probability of the next roll coming up $6$? You might say $1/6$. But it was never declared to be a fair die. In ...
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1answer
178 views

Triangle Distribution, How to find Upper Bound ? if you have median and lower bound

if the lowerbound is 3 and Median is 9, How do I calculate the Upper Bound ? I have been told x is drawn from a symmetric triangle distribution. Im not sure which value to use(I have to sub it into a ...
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29 views

Hidden Markov Model Confidence Interval (preferably in MATLAB)

I'm trying to uncover the transition parameters of data of a hidden Markov Model using MATLAB. Using the built in hmmtrain function, I can estimate the parameters quite well (I already know what they ...
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108 views

Poisson distribution-Queueing theory

Vehicles arrive at a junction, in order to swing left, create a line queue ( tail) . The number of vehicle follow Poisson distribution. The length of cycle for the traffic light (for left turns ) is 1 ...
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17 views

Trimmed mean is translation invariant Statistics.

I want to know why trimmed mean is a translation invariant statistics? I know that if $\mu$ be a measure on $(\mathbb R^n, \mathcal B)$ then it is called translation invariant iff $\forall x \in ...
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30 views

Average Length Of Queue

If I have a queue that randomly queues and dequeues, then if each time the queue grows and shrinks I add the queue size to an integer, then at the end, divide by the number of times the queue size ...
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2answers
42 views

Normal Distribution in

I am so confused with this problem: The middle 95% of adults have an IQ between 60 and 140. Assume that IQ for adults is normally distributed. a. What is the average IQ for adults? The standard ...
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24 views

How to show test is UMP when not of the NP-Lemma test form?

I am testing the hypothesis H: $\theta = \theta_0 $ vs the alternative A: $\theta = \theta_1$ or $\theta_2$ for the random variable X which takes on integer values 1 through 6 with varying ...
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2answers
138 views

maximum likelihood - estimate $\sigma^2$ in $N(0,\sigma^2)$

Prove, using maximum likelihood, the estimation of $\sigma^2$ where $X$ is $N(0,\sigma^2)$ There is no real statistics involved, just algebra and finding partial derivatives, so I tagged it algebra ...
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1answer
24 views

Prove easy sum relationship

I want to understand the proof of how to estimate variance from the Normal distr, using the method of moments. It boils down to simple algebra. I can't understand why I tried to prove it like this ...
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1answer
69 views

Normal Distribution

Would greatly appreciate any help on this homework question, I will post my answers to parts a) and b) underneath as well but I don't think they are correct.Thanks! a) Take 10 different samples of ...
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40 views

Let X be exponentially distributed with mean 1. Let Y|X=x be exponential with mean x. You are interesting in estimating P($XY\leq3$).

I have some difficulties with homework. And I would be glad if you help me. Let X be exponentially distributed with mean 1. Let Y|X=x be exponential with mean x. You are interesting in estimating ...
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1answer
37 views

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$?

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$? Standardizing, $P(\frac{X - 5}{\sigma} < \frac{-1 - 5}{\sigma}) = 0.1587$ $P(Z < ...
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1answer
35 views

Measure Accuracy using Statistical Tool

I want to measure the accuracy of my GPS Receiver module. The real coordinates are obtained from the Google Maps, and the actual received coordinates are the ones that the GPS receiver received. I ...
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1answer
22 views

poisson distribution and the cdf

$Y (t)$ is the number of events occurring in $[0,1]$ where for each $t> 0$, $Y (t)~\sim\operatorname{Poi} (\lambda t)$ and $X$ measures the time taken for the $r$th event to occur. Am I right in ...
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35 views

calculating difference in % using standard and safe methods. Please help.

In the plant physiology lab you sow 30 rye grass grains in each of two filter paper lined petri dishes. Into one dish 7.0ml of water is added and in the other dish 7.0ml of a 10 mg/l Coumarin solution ...
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1answer
35 views

Expectation of a Uniform PDF

How do I find the expectation of the following pdf? $f(x,y) = 1/\pi r^2$ , where $x^2+y^2 \leq r^2$ I've tried to integrate it on the bounds $-\sqrt{1-x^2}$ and $\sqrt{1-x^2}$ for $\int ...
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15 views

Constructing confidence variances for std and variance samples.

A) Construct the 90% confidence interval for the standard deviation for the dark treatment. B) Construct the 95% confidence interval for the variance for the sunlight treatment. And that what I ...
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44 views

Bivariate distribution

We are given two independent random variables, first is $X$ which follows a normal distribution $(1,1)$ and second is $Y$ which also follows a normal distribution $(2,1)$. We are given $Z=X+Y$ and ...
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1answer
42 views

Sum of exponentials, distrubution [closed]

The sum of $n$ number of independent $X_i$ where each $X_i$ follows $$\exp(\lambda_i),$$ then $$Y=\min [ X_1, \dots , X_n]$$ follows ... what distrubution? Is it $exp( \Sigma \lambda_i)$?
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1answer
116 views

log likelihood function of a cauchy distribution

What is the log likelihood function of a random varible x with cauchy distribution (0,1)? I've tried to work it out. I think its $\log (1+x)^2$. Is that correct?
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2answers
216 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
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2answers
38 views

Estimating two constants from fitted equation

I have an equation $A=\alpha[1-\exp(-\beta\cdot B)]$ where $A$ and $B$ are known 20x1 column vectors and $\alpha$ and $\beta$ are unknown constants. I'm probably missing something really simple but ...
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How to compute a “weather” score for repeated trials over time, with a bias for recency

I'm doing some analysis on repeated software tests that can either pass or fail. For each test, I've collected a set of historical (pass/fail, timestamp) records, and I'd like to compute a score to ...
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1answer
28 views

Confidence interval of a binomial distribution?

I'm trying to use the Clopper Pearson Interval for a binomial distribution, but am not sure how to find it, despite having the formula: What does this mean ...
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42 views

Yule Walker equations

The Yule-Walker equations relate the auto covariance of a random signal to the autoregressive (AR) model parameters. They can be used to estimate AR models from data by first estimating the auto ...
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22 views

How can the confidence interval of a multinomial distribution be found?

The confidence interval of a binomial distribution can be found using Normal Approximation Interval, Clopper-Pearson or Wilson Score Interval (along with others). But what can we use if the ...
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38 views

How to convert a histogram into a failure rate distribution?

I have a histogram in which the frequency of failures are charted in the y-axis, the bins in the x-axis are time pockets in hours. This is from my historical data. The y-axis is the number of failures ...
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1answer
111 views

Hypothesis testing question

The article “Statistical evidence of discrimination” (J. Ameri. Stat. Assoc., 1982, 773-83) discussed the court case of Swain v Alabama (1965), in which it was alleged that there was discrimination ...
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1answer
62 views

Largest Number Drawn - Why are These Approaches Not Equivalent?

Here's the question: Four numbers are drawn at random from a box of ten numbers 0, 1, ..., 9. Find the probability that the largest number drawn is a six if the draws are made with replacement. The ...
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1answer
37 views

How can I $P(X+Y\neq0)$ [closed]

let $X\sim N(0,1)$ ,$Z$ be independent of $X$ and $P[Z=1]=1-P[Z=-1]=p$. if $Y=ZX$ how can calculate $P(X+Y\neq0)$