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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Simple math for Amoeba's survival type problems

Here's an example but it can be different too, like about people and their children, etc. http://www.cut-the-knot.org/Probability/AmoebaSurvival.shtml A population starts with a single amoeba. For ...
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1answer
34 views

Central Limit Theorem and adjusted standard deviation

Per capita consumption of Spam in the US is Normally distributed with a mean of 2.3 pounds and a standard deviation of 0.9 pounds. Random sample of 20 people are chosen. Use the Central Limit Theorem ...
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40 views

Finding a likelihood function given binary observed data

I'm having trouble really understanding the terms used for this homework question, and what I am actually supposed to be doing, given the actual data for the problem. Below is the problem: Suppose we ...
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1answer
17 views

Binomial and probability

History indicates that 80% of all McDonald's customers order French fries with their meal purchases. During one afternoon at a slow restaurant, 15 people order meals. Assuming that all criteria for a ...
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207 views

How to calculate Fisher Information (FI) matrix for Multivariate Normal Distribution (MN)

Below is the gradient (score) of the MN log likelihood function L for n=1 observation. I originally attempted to calculate the Hessian matrix but ran into difficulty calculating 2nd order derivatives ...
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16 views

Justification for Inverse-Variance Weighting?

In Inverse-Variance Weighting, $1/\sigma^2$ is defined as the weight of the random variable. Why is this?
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34 views

Z-Score and stats

At 45 mph, the braking distance of an automobile has a mean of 50 feet and a standard deviation of 3.5 feet. Find the braking distance that corresponds to a z-score of 2.0. This is the boundary of an ...
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37 views

How can I solve for X in a sum function?

I have this inequality constrain, let's call it (*) $$\frac{1}{C}\sum_{i=1}^{n} X_{i}Y_{i} < Z$$ (forgive me if it should be $i=0$ and $n-1$, this is my first question here and I'm very rusty) ...
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2answers
61 views

How to compute probability related to a difference of two random variables

I am studying Joint Probability Distributions and Random Samples. I have a function for a probability distribution, defined as: $ f(x, y) = K(x^2 + y^2)~~~~~~~~~ 20 \leq x \leq 30, ~~~20 \leq y ...
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15 views

Mean and standard deviation of a set of measurements with error that is only partially correct

There is a number of measurements $\mu_i$ with standard deviation $\sigma_i$. The given $\sigma_i$ are only the lower bound of the true error of each measurement, there is a missing and unknown ...
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1answer
41 views

Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
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1answer
51 views

Calculating probability distribution under given constraints

I recently asked a question about the construction of a random variable under given constraints (see: Construct a random variable under given constraints). The only answer to my question suggested a ...
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21 views

Can I have the following random matrix?

Let $A \in \mathbb{R}^{n \times n}$ a random matrix. Assume $E[A] = I$ and that all $A_{ij}$ are independent. Now let $U \Sigma V^{\top} = A$ be the SVD of A. Let $A'$ be the result of thin-SVD, ...
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1answer
82 views

Maximum bound on variance of sample proportion

A poll was conducted to gauge the attitudes of residents to moose culling. Let p^ be the proportion who approve of the culling. For a given sample size n, what is the maximum possible value of ...
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1answer
45 views

Expectation for number of students who miss true value of $\beta_1$

I'm having a hard time understanding how to complete a part of a question from an assignment. The question states: Suppose we have 103 students in our class. Suppose all students collect a random ...
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1answer
19 views

Verification of linear combinations of a normal distribution

A machined part consists of 5 independent components connected end-to-end. Two of these have lengths $N(37.0, 0.49)$, and three of these have lengths $N(24.0, 0.09)$. All measurements are in mm. What ...
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69 views

Reasoning for confidence interval

Suppose $$X_1,\dots,X_{20} \sim f_X(x;\beta)$$ where $$f_X(x;\beta) = \frac{1}{\beta} e^{-\frac{x}{\beta}},\quad x>0;\beta>0$$ It can shown that ("details omitted") $$P(0.52 \bar{X} \leq \beta ...
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1answer
127 views

Compound of Exponential and Inverse Gamma Parameter

I am trying the prove the following: Show that an exponential random variable such that the inverse of the parameter is gamma-distributed is Pareto-distributed. More precisely, show that if $$X | M = ...
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2answers
97 views

Manipulating continous random variables

Question: Suppose X~ Uniform(0,1) and $Y=x^3$.What is the pdf for Y? my attempt: so, X follows a uniform distribution with minimum of 0 and max 1,so $f_X(x)=1.$ Using the definition that the pdf of ...
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1answer
115 views

Application of Compound Poisson Process

I am trying to solve the following application problem: The life T (hours) of the lightbulb in an overhead projector follows an Exp(10)-distribution. During a normal week it is used a Po(12)- ...
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1answer
250 views

Find The Z-score with a percentile

Relatively basic question here .. A Company wants to give gifts to the top 5% of customers. If the mean spend per customer is $ 135 - with a standard deviation of 55 - what balance should the ...
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27 views

Confidence interval for binomial distribution

Let $X_1,...,X_n$ be a simple sample of Bernoulli distribution with success parameter $\theta \in (0,1) $ I have to find the exact (not asymptotic) confidence interval for unknown parameter $\theta $ ...
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1answer
79 views

2-way ANOVA degrees of freedom proof

Consider the 2 way ANOVA model: $y_{i j} = \mu + l_j + \alpha_i + e_{i j}$, $e_{i j} \sim \mathcal{N} (0, \sigma^2)$ where $i=1\dots m, j=1\dots n$, how do you prove that the residual degrees of ...
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2answers
66 views

Calculating the sample standard deviation

I've been preparing for an exam tomorrow and was doing a past paper when it appears the mark scheme has an answer that I just cannot fathom. The exact question is to "find a 95% confidence interval ...
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2answers
133 views

Probability of a letter being in a four letter word.

Say you have a word with 4 available slots, take the word "bake" as an example. You want to compute the probability of a letter (from the English alphabet) being in a slot of the word. So what's the ...
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39 views

Distribution from moment generating function [closed]

Moment generating function for $ X ~ (\vec{\mu}, \Sigma) $ is of form $ M_x(t) = exp( t^T\vec{\mu}+\frac{1}{2}t^T\Sigma t)$ The random variable $X = [T_1, T_2]^T$ has moment generating function ...
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21 views

Propagating uncertainties in Gaussian fit

I'm doing an analysis where I have a set of random variables with some known uncertainties (the uncertainties are different for each random variable). The random variable is roughly Gaussian ...
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2answers
62 views

Construct a random variable under given constraints

In preparation for a probability examination, I am working on the following problem. Problem A box contains three white balls and ten black balls. Balls are drawn without replacement until all the ...
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2answers
41 views

General method to find exact confidence interval?

Say we have a beta random variable with pdf, $f_X (x) = e^{-x/\theta}/\theta$ for positive $x $. Find the exact confidence interval of $\theta$ with 95% confidence. A solution to this says ...
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52 views

Significance test for the mean squared error skill score (MSESS)

i am looking for a test statistic whether a calculated value of the mean squared error skill score is significantly different from zero. The MSESS is defined as: $MSESS = 1 - \frac{MSE_H}{MSE_O}$ ...
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4answers
52 views

How to determine the number removed from the list [duplicate]

One number is removed from a set of integers from 1 to n,the average of the remaining numbers is $\large{\frac{163}{4}}$. Which number was removed? I tried to find the mean of ...
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1answer
50 views

Hypothesis Testing - Review Question With Push-up Amount

As the title says, this is a review question. However, I'm having trouble solving it, particularly at the final part when I reject/accept the null hypothesis. The question is: Twenty years ago, ...
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3answers
193 views

Probability question for horse racing

I know there is quite a few of these, but for some reason I have so much trouble understanding probability properly, so was hoping to get some help. Basically there are 8 horses racing, assuming they ...
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1answer
305 views

Calculating test statistic of a poisson distribution

I'm in an intro stats class, and I'm wondering how I can argue or prove the question below regarding sample size and poisson distribution. Suppose a company which produces fire alarms has claimed ...
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1answer
41 views

Find the sample space for a committee of two chosen from Alice

Find the sample space for a committee of two chosen from Alice (A), Bill (B), Carol (C), and David (D). {(A, B), (A, D), (B, C), (C, D)} {(A, B), (C, D {(A, B), (A, C), (A, D), (B, C), (B, D), (C, ...
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1answer
36 views

Non-integer $n$ in sample size problem

Setup Consider a sample size determination problem with the maximization of expected utility approach (as in Lindley 1997). Let $\theta$ be the state, $x=(x_1,\dots,x_n)$ a sequence of $n$ iid ...
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1answer
55 views

Intuitive explanation for variance as expectation of square minus square of expectation

$$\text{var}(X)=\text{E}(X^2)−(\text{E}(X))^2$$ I am able to follow the proof of the above from the basic definition of the variance, and mathematically it seems fine but I cannot seem to grasp ...
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2answers
47 views

Skewness in a graph of data

It seems counter-intuitive to me that a graph that is skewed right actually has more data values on the left side (and vice-versa), like in this image: I don't get how the term "skewed right" ...
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1answer
427 views

Finding a p-value based on the Poisson distribution

I'm having trouble understanding what I need to do to solve this problem for homework: Suppose a company which produces fire alarms has claimed that the fire alarms make only one false alarm per ...
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1answer
59 views

Ratios of conditional probabilities given a contingency table

I'm having issues trying to understand what I need to use in order to get a correct answer for this homework problem. On top of that, I just really want to understand exactly what I should be doing ...
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23 views

Chernoff Bound: Prove that $P[u \geq \alpha] \leq (e^{-s\alpha} U(s))^N$.

Let $u_1, ..., u_N$ be random variables, and let $u = \frac{1}{N} \sum_{n=1}^N u_n$. If $U(s) = E_{u_n}(e^{su_n})$ (for any $n$), prove that $P[u \geq \alpha] \leq (e^{-s\alpha} U(s))^N$. $s > 0$ ...
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1answer
47 views

Binomial cumulative distribution calculation

I have exercise like this: Given probability of success of 0.8 what's probability that in 1000 trials there was more than 800 successes? So it's quite simple exercise: It's binomial distribution ...
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1answer
32 views

Confidence interval and symmetric distribution

Let $X_1,...,X_n$ random sample of $X$~$U[\theta-\frac{1}{2};\theta+\frac{1}{2}]$.Consider $[X_{(1)};X_{(n)}]$ a confidence interval for $\theta$. Find their confidence level and show that ...
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38 views

Geometric Mean of Random Variables

I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and ...
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1answer
88 views

Probability about confidence interval

Let $X_1,...X_n$ be iid $N(\theta,1)$. A 95% confidence interval for $\theta$ is $\overline{X}\pm\frac{1.96}{\sqrt{n}}$.Let p denote the probability that an additional independent ...
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1answer
74 views

Margin of error of poll. [closed]

A news agency publishes results of a recent poll. It reports that candidate A leads candidate B by 10% because 45% of the poll participants supported Ms. A whereas only 35% supported Mr. B. What ...
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1answer
29 views

Probability/Statistics help? [closed]

In a population, height of females are normally distributed with mean $\mu_1 = 162\mathrm{cm}$ and standard deviation $\sigma_1=6\mathrm{cm}$. Heights of males are normally distributed with mean ...
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21 views

Support of a Random Variable and Linear Subspaces in $\mathbb{R}^d$

I am reading a Text about Single Index Models where a theorem is given for the identification in case all covariates are continuous. The theorem states these four conditions: $G$ is differentiable ...
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30 views

Pitman Asymptotic Relative Efficiency

Let $T_1$ and $T_2$ are two test for testing the same $H_0$ versus $H_1$. Let $t_1$ and $t_2$ are test statistic correspond to $T_1$ and $T_2$, respectively. If $t_1$ and $t_2$ have noncentral chi ...
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1answer
109 views

Central Limit Theorem: Finding the probability that a sample proportion will differ from the population by more than a given amount

Suppose 48% of the registered voters in a county are Republican. If a sample of 586 voters is selected, what is the probability that the sample proportion of Republicans will be greater than 42%? ...