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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Expected width of the confidence interval for the variance

Suppose that $X_1,\ldots,X_n$ are iid normal random variables. The confidence interval for the variance with the $1-\alpha$ confidence level is given by $$ \biggl[\frac{\sum_{i=1}^n(X_i-\bar ...
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1answer
19 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
43 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
30 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
20 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
22 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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0answers
13 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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0answers
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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0answers
12 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
11 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
23 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
14 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
36 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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0answers
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
46 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
37 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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0answers
18 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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0answers
12 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
3
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1answer
36 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
11 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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0answers
9 views

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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6answers
51 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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0answers
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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0answers
14 views

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

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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0answers
14 views

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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0answers
13 views

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 ...
2
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1answer
20 views

Limiting Distribution $\Delta-$method

Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$. What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. ...
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1answer
21 views

Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
3
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1answer
54 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
2
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1answer
25 views

Likelihood function for a distribution with both discrete and continuous components

Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood ...
2
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2answers
36 views

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator?

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator? Here's what I tried, is this right? $$ \begin{align} ...
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0answers
6 views

An applied example of Moment Generating Function in R? Or other software

I'm trying to understand MGF, I get the theory but I'd like to find an example I can relate using a software, like R studio or Matlab. Any example of one would be really appreciated!
3
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1answer
19 views

Median of Medians

Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ? My first thought was that this wasn't true so I tried to find ...
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0answers
25 views

Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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0answers
10 views

How to find variance of weighted sum in terms of inputs and weights ? Variance of inputs and weights are given. [on hold]

Let $X = [x_1, x_2, ... , x_M ] $ and $ W = [w_1, w_2, ... , w_N] $ $y_i = {\sum_{j=1}^N}{\sum_{i=1}^{M} {x_iw_j}}$ Variance of $X$ and $W$ are given. Can we find the variance of $Y$ in terms of ...
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1answer
32 views

If $Y = (\mathcal{N}(\mu_1,\sigma_1^2) + \mathcal{N}(\mu_2,\sigma_2^2))^2$, what is $\Pr(Y>\mathrm{E}[Y])$?

Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$? ...
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1answer
33 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
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2answers
28 views

Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
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1answer
43 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
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2answers
14 views

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
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0answers
17 views

probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
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25 views

Which distribution would be the most appropriate?

What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns? e) The size of an automobile insurance claim I'm thinking that the distribution ...
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1answer
26 views

An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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2answers
23 views

Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
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0answers
21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...