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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Use Maximum Likelihood Estimation to guess which dice got selected

We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the ...
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2answers
72 views

Percentage greater than 2 standard deviations from the mean

A question reads: "The weights of $910$ young deer tagged and weighed in a research study are normally distributed with a mean of $86$ pounds and a standard deviation of $2.5$ pounds." Approximately ...
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8 views

Maximum likelihood estimate lies outside the paramater space

Say if I have a model where I impose the restriction that $\hat{\theta} \in (0,1)$, and I calculate the MLE to $\not\in (0,1)$, does this mean my model is incorrect, for this parameter restriction?
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1answer
28 views

When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?

The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity ...
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1answer
22 views

Given probability distribution $f(x)=2-bx$ find $b$ and range for $x$

Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we ...
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1answer
14 views

Confusion regarding the weak law of large numbers

I can intuitively understand that as I take more samples from a random variable $X$ (gaussian distribution), the mean would approach $E(X)$. But what I don't get is if I look at it mathematically. ...
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3answers
44 views

What is the difference between 10% and $\frac{1}{10}$

In a national competition , ech school had to choose 10% of students who participated in the competition . So my question is , why they didn't asked for $\frac{1}{10}$ of students who participated ? ...
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26 views

Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone ...
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11 views

Is Pearson's chi squared test the right method?

I have a sample of n=1000. The sample covers cars being brought in for service after one year of ownership in my country. For each car, I know which defects it had when it was brought in. I'm trying ...
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1answer
26 views

Why is $E(X_2|X_1) = X_1$?

From textbook: $f(x_1, x_2) = 2 e^{-x_2/x_1},$ where $ 0 < x_1 < 1$, and $ x_2 > 0.$ The marginal is $f(x_1) = 2x_1$, and accordingly $$f(x_2|x_1) = \frac{1}{x_1}e^{-x_2/x_1}.$$ My ...
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48 views

limiting value of an expectaion of a sum of random variables [on hold]

Let $X_1,X_2,X_3,\dotsc$ be a sequence of i.i.d. $N(\mu, 1)$ random variables. Then, $$\lim_{n\to\infty}\frac{\sqrt \pi}{2n}\sum_{i=1}^n E\left(|X_i-\mu|\right)$$ is equal to ____________. ...
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17 views

Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator.

If $Y_1, Y_2, \ldots , Y_n$ denote a random sample from an exponential distribution with mean $θ$, then $E(Y_i)=θ$ and $V(Y_i)=θ^2$. Thus, $E(\bar Y)=θ$ and $V(\bar Y)=θ^2/n$, or $σ_Y=θ/\sqrt{n}$. ...
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28 views

show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{exponential}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. ...
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1answer
23 views

Why does $E(C\cdot \epsilon\; \vert\; C\cdot X) = E(C\cdot \epsilon\; \vert\; X)$?

Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$ This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he ...
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2answers
34 views

What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
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1answer
24 views

Troubles With The Beginning

The following is the question I'm having a bit of troubles starting: Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the ...
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17 views

Finding joint distribution for the following

I am trying to do the following problem Suppose that $X_1,...,X_n\stackrel{iid}\sim N(0,1)$. Define $$\bar{X}_k=\frac{1}{k-1}\sum_{i=1}^{k-1}X_i,\,\,\,\,\,\,\text{for }k=2,3,.....,n $$ (i) What is ...
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1answer
14 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
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Is a random variate a functional?

A https://en.wikipedia.org/wiki/Random_variate is a particular outcome of a random variable. I was wondering what is meant in the Wikipedia article with "a random variate is corresponding to a ...
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10 views

Statistics: Attributable Risk.

$AR$ = Attributable Risk Suppose that we were determining the effects of smoking on heart disease. We found that among smokers in a certain age range, a proportion of $.4532$ with heart disease and ...
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1answer
15 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
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2answers
42 views

Methods for calculating the mean and variance of a distribution created from the addition of two normally distributed quantities

I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables. ...
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1answer
25 views

Calculate $P[A,B,C]$ from $P[A,B]$ and $P[B,C]$

I have 3 (not independent) events $A, B, C$ and I know everything about how any two of them correlate. For example, I know: $$ P[A], P[B], P[C], P[A,B], P[A,C], P[B,C], P[A|B], P[A|C], P[B|C], ...
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2answers
46 views

Find probabilty

I have this table of information: Probabilities: \begin{array}{c|c} .919 & ????\\\hline ???? & .274 \end{array} How do I find the probabilities of the question marks? I thought each row ...
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1answer
33 views

Two candidates, A & B, are running for president. What is the probability that candidate A beats candidate B?

Candidate A has already garnered 80 votes. Candidate B has already garnered 50 votes. The number of votes a candidate must have in order to win the election is 115. The votes of 5 states are still ...
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7 views

How do you show that {ηt} is a white-noise series? [on hold]

Given that $a_t^2 = σ_t^2(1+ε_t^2-1)=σ_t^2+σ_t^2(ε_t^2-1)=σ_t^2+η_t$ Assume that the variance of $η_i$ is constant for all t.
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1answer
23 views

Proof of independence of $\bar{X}$ and $S^2$: Trouble understanding $n$-dimensional Jacobian Result

I'm trying to work through the proof given here that the sample mean and sample variance of a random sample $X_1, X_2, ..., X_n \sim N(\mu,\sigma^2)$ are independent. The part I can't seem to follow ...
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1answer
26 views

Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
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15 views

Cramer-Blackwell estimator for uniform distribution.

I've got two estimators of parameter $\alpha$ in the distribution $X=X_1,...,X_n$, where $X_i$s are i.i.d. uniform random variables on the interval of $(0,\alpha)$. These two estimators are: ...
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5 views

Determining forecast error of realtime prediction of binary outcomes [migrated]

Given datasets consisting of a daily prediction and confidence percentage for each of a small number of binary outcomes, what is the proper way to calculate the forecast error of each series and of ...
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13 views

Convexity for Hoeffding's Inequality

We consider a r.v. $X$ that satisfies $0 \leq X \leq 1$ a.s. and a sample of $n$ i.i.d. random variables $X_1,\dots, X_n$ with the same distribution as $X$. We denote by $\mu= E[X]$ and we let ...
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1answer
16 views

Estimating confidence Interval for unknown Variance, Normal distribution

I've been stuck with this question for a while: I've learnt how to use t-distribution to estimate CIs for an unknown variance, but I'm unsure how that applies to this situation. Any help would be ...
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1answer
25 views

Consider a probability distribution on the numbers $0, 1, 2, …, 20$ with probabilities given $P(k) = \frac a{2^k}$ . [duplicate]

Find $a$ so that the sum of all probabilities is $1$, and hence $P(k)$ is a well-defined formula for a probability distribution.
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1answer
16 views

Simultaneous Confidence Interval

I don't understand the example here. If we have that $\alpha=0.04$, $k=2$ then $[L_1,U_1]$ is a $1-\frac{0.04}{2}=0.98$ confidence interval for $\theta_1$ which is correct, however $[L_2,U_2]$ ...
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3answers
40 views

Help with finding the probability of this exam question

I need help with solving one of the questions the teacher gave us to prepare for an upcoming exam. I tried solving it but with no luck. Here is the question: On one shelf there are 5 hardcover ...
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13 views

bell curve (emperical formulas)

I have a frequency (normal distribution) graph and I am asked to draw $6$ lines to show the empirical rule I know the lines must show ($99.7$%, $95$%, and $68$%) but still I do not know how to draw ...
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0answers
50 views

Proof of the existence of a reversible stationary distribution

$p$ is a finite Markov chain where $p(i,j)>0$ for all $i,j$. Prove a reversible stationary distribution exists for $p$ if $p(i,j)p(j,k)p(k,i)=p(i,k)p(k,j)p(j,i)$ for all $i,j,k$ This question is ...
2
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1answer
38 views

If X and Z are independent and Y and Z are independent random variables, is cov(XY, Z) = 0?

Let $X$, $Y,$ and $Z$ be random variables. (There are no restrictions on these variables, but you may assume that these are continuous random variables if you want.) Suppose that $X$ and $Z$ are ...
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19 views

Consider a probability distribution on the numbers 0, 1, 2, …, 20 with probabilities given P(k) = a( 1/ 2 ) k . [on hold]

Find a so that the sum of all probabilities is 1, and hence P(k) is a legitimate formula for a probability distribution. ANd the probs distrubtion goes from 1 to2.. How do I start this problem
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1answer
38 views

Determining an upper bound

I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$ I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion. I have used the taylor expansion for ...
2
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1answer
14 views

Moment generating function of sample mean of bernoulli random variables

Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} ...
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What is the Bernoulli class conditional distribution?

What is the Bernoulli class conditional distribution? I am trying to implement a procedure for computing a naive Bayes classifier for binary features with a Bernoulli class conditional distribution. ...
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53 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
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9 views

Sampling Distributions. Statistics [on hold]

I'm stuck in this problem: Problem Picture I did the literal a and b, but the rest of them I don´t understand, the reason why.
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29 views

probability that a customer who purchases up to $5$ songs from $4$ music genres prefers jazz and buys at least $3$ songs

Customers can choose from $4$ music genres: jazz, rock, new age, country; and can purchase up to $5$ songs. The events are: $A =$ customer prefers jazz and buys at least $3$ songs $B =$ the customer ...
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Why does ergodicity not neccesarily imply ergodic for the mean?

I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 ...
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1answer
12 views

Hypothesis test of a sample that contains both male and female.

If I want to do a hypothesis test of a sample that contain $49$ women and $51$ men. The hypothesis test is only regarding the women which has a given sample standard deviation of $12,032$ and sample ...
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Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
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26 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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1answer
46 views

References for information theoretic statistical tools

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are necessitated in my ...