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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How do I calculate probabilities from a probability mass function?

If I have a PMF like so Are the following correct? $P(5 \leq x < 7) = 0.9$ $\quad (P(1) + P(3) + P(5))$ $P(1 < x \leq 5) = 0.5$ $\quad (P(3) + P(5))$ $P(5 < x \leq 7) = 0.1$ $\quad ...
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20 views

Travel- Find the average time given a standard deviation

It is known that the travel time from A location in the northern part of Sydney to the B follows a normal distribution with standard deviation 9 minutes. In 200 days was noted the time to come that ...
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106 views

Empirical Kullback-Leibler divergence of two time series

I have an two vectors (time series) with the same length (1200 elements) $x$ and $y$. Further both time series are stationary. I don't know the theoretical distribution of $x$ and $y$. I would like to ...
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156 views

Confidence Interval problem

Which of the following will result in a wider confidence interval? Check all that apply. ...
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30 views

Is the sum of two exponential distributed random no. is also exponential random number?

I am working on statistics after long time. Struggling with the basics. Is the sum of two exponential distributed random no. is also exponential random number?
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18 views

Finding Moment Generating Function example

Given probability mass function $$P(X=x)=k\binom{n}{x}$$,$x=0,1,...,n$ Where $k$ is a constant. Find the mgf? It is an easy question but i am wondering about that constant, i find the solution is ...
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31 views

Tranformation of random variables

Let $X$ have the p.d.f $f(x)= e^{-x}$, $ x > 0$, $0 $ otherwise,find the pdf of $Y = X^2$ and space range $Y$. I use the change of variable formula The inverse is $${x} = \sqrt{{y}} $$ The ...
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24 views

Probability of not getting any left handed people in study

The question says about 11% of the world population is left handed. If you pick a random sample of ten people, what is the probability of getting no left handed people?
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34 views

Please show that $f(\beta_0,\beta_1)=\log(1+\operatorname{exp}(-y_1(\beta_0+\beta_1 x_1)))+\log(1+\operatorname{exp}(-y_2(\beta_0+\beta_1 x_2)))$

I would like to show that the following result is indeed true. I am very new with this subject, so I ask for a hint to get me started please. Please show that ...
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16 views

Inferential Statistics problem about a poll and samples

The wording of the problem is a bit odd, but I'll try to keep true to the translation: ...
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19 views

Distribution of ratio of 2 points drawn from normal distribution?

Let's say we have a known normal distribution $N(\mu,\sigma^2)$. I now draw 2 points $p1$ and $p2$ randomly from this Gaussian distribution for every observation, and repeat this process large number ...
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17 views

Find A and B for a continuous random variable with the following density

Suppose $X$ is a (continuous) random variable with density $$f_X(x) = \begin{cases} 0 & \mbox{if }x ≤ 0 \\[0.25ex] (A/x^2) & \mbox{if }0 < x < 1 \\[0.25ex] (B/x^2) &\mbox{if }x ...
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18 views

If $X$ is binomial$(n,p)$, find unbiased estimator for $p^2$

So I know I'm looking to find a function $\delta(X)$ so that $E\left[\delta(X)\right] = p^2$. Let me venture a guess that $\delta = \hat{p}^2 = \frac{x^2}{n^2}$. Then: $$E\left[\hat{p}^2 \right] = ...
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26 views

In what sense is the limit meant in von Mises approach to probability

In a frequentistic approach to probability as proposed by von Mieses and otheres, the term "propability" $P(A)$ of an event $A$ is defined as $$ P(A) = \lim_{n\to\infty} \frac{n_A}{n} $$ where ...
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32 views

Prove using weak law of large numbers

Can someone help me out? Let $X_1,X_2,...,X_n\sim N(0,1)$ with $$ \mathbb{C}\textrm{ov}(X_{i},X_{j})=\begin{cases} \begin{array}{ccc} \rho & \textrm{if} & \left|i-j\right|=1\\ 0 ...
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1answer
24 views

Probability of a Specific Result

I am trying to validate a small Monte Carlo that I am running: A small bucket contains seven ping pong balls numbered $1$ through $7$. A ball is drawn at random from the bucket, the number recorded, ...
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14 views

Is there a difference between density and distribution in statistics?

I'm trying to find the density of the random variable Y=X^2 and i've been given an example but it shows how to find the distribution of a random variable. So I was just wondering if they were the same ...
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12 views

definition of conditional probability $(p_i|\pi_k)$ and Tsallis entropy

Let $\Omega$ be a set of $W$ possible outcomes of an experiment with probability assignments $p_i$ and thus $\sum_{i=1}^{W}p_i=1$. Now, let's divide $\Omega$ into $K$ non-intersecting subsets each ...
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358 views

Two envelope paradox, instead with amounts distributed uniformly.

There are two envelopes, each of which has a check for a Unif(0, 1) amount of money, measured in thousands of dollars. The amounts in the two envelopes are independent. You get to choose an envelope ...
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66 views

Statistics (Faster Win = Higher Certainty of Superiority?)

I have a question about statistics and I'm not quite sure how to explain it concisely so please bear with me. I am ranking characters in a video game called Mugen (2D fighting game) by order of ...
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210 views

AP Statistics Chapter 6 MC Review (TOPIC: Combining Normal Random Variables)

“Insert tab A into slot B” is something you might read in the assembly instructions for pre-fabricated bookshelves. Suppose that tab A varies in size according to a Normal distribution with a ...
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83 views

Solve the mgf and expected value of a discrete random variable

This problem is from Casella and Berger A distribution cannot be uniquely determined by a finite collection of moments, as this example from Romano and Siegel (1986) shows. Let X have the normal ...
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13 views

MSE risks for estimators of $\sigma^2$

Compare the MSE risks for the two common estimators of $\sigma^2$ in random sampling from the normal with both parameters unknown. (The one estimator has $n-1$ in the denominator, and the other ...
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Can't figure out problem using joint density with bivariate normal distribution.

Not even sure where to start with part (b) for the problem below. For part (a) assuming the worker knows her own skill level and the prevailing wage, I got: y1 > y 0, or S1 - S0 > ln(w0/w1) for ...
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35 views

Total accumulated risk

I'm a big fan of Minesweeper. Like other fans, I know that probabilities are important in this game. So, for more fun, I play on this player : http://mrgris.com/projects/minesweepr/demo/player/. ...
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33 views

P(A|B') from P(A|B)

I know $P(A|B)$. From this, is there a formula to get $P(A|B^c)$ where $B^c$ is "not $B$". It seems like there would be some connection similar to $P(A^c|B)=1-P(A|B)$.
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40 views

Covariance dependent Binomial variables

Suppose $Y \sim \mathrm{Bin}(n,p)$ given $\Theta=(p,q)$ and $Z \sim \mathrm{Bin}(y,q)$ given $Y=y$ and $\Theta=(p,q)$. Now I want to determine the variance of $Z-Y$, but I don't know how. I know ...
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1answer
20 views

Time Series with increasing dimension

I am stuck with the following problem from research. I am not sure how to model this situation. I have a vector time series whose dimension increases with time, $t$. Specifically, let ...
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2answers
85 views

Why is $R^2=\rho^2$

Considering $y_i=\beta_1+\beta_2x_i+\epsilon_i$ $\bar y_i=\hat\beta_1+\hat\beta_2\bar x_i+\bar\epsilon_i$ a linear equation of least square used when it seems that there is a like between two data, ...
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356 views

Prove that the Expected value of Y bar ^2 = µ^2

I'm trying to show whether or not $\bar(Y^2)$ = $\\µ^2$ Or the mean of the sample squared) is a biased or unbiased estimator of the population mean squared. I can prove that Ybar is an unbiased ...
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25 views

Upper bound on an expectation

I'm looking for an upper bound on $E(X^k)$ where $X$ is a random variable with $E(X)=1$. How can I go about doing this?
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21 views

Formula for the n'th order correlator $\langle \Gamma(t_1)\Gamma(t_2)…\Gamma(t_n)\rangle$ of Gaussian white-noise?

Is there a closed form formula for the n'th order correlator $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\langle \Gamma(t_1)\Gamma(t_2)...\Gamma(t_n)\rangle$, of Gaussian white-noise ...
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2answers
61 views

Least Square Approximation for Exponential Functions

I'm a little confused on how to approach the following problem. Use the least square method to fit a curve of the form $y=a\cdot b^x$ to a collection of $n$ data points $(x_1,y_1),...,(x_n,y_n)$ ...
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67 views

Ranking System on Multiple Choice Tests

Suppose we have a Multiple Choice Test with the following characteristics: There are M available multiple choice Questions on the system There are N users participating Each user replies to K ...
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1answer
29 views

Area of Probability Density Functions vs. Probability Mass Functions [closed]

Do both of these functions have a total area of 1?
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1answer
19 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
3
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2answers
43 views

MSE of an estimator as sum of bias and variance

I am reading that how the MSE of an estimator $\hat{\theta}$ of $\theta$ can be expressed as $E(\hat{\theta} - \theta)^2$. Then this can be further simplified to $ (E[\hat{\theta}] - \theta)^2 + ...
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1answer
18 views

Joint discrete probabilities

I want to find out the expected value E[ABC]. When B takes on 0, A takes on 0.5 When C takes on 0, A takes on 0.1 B and C cannot both take on 0 A takes on two values: 0.5 - 80% of the time 0.1 - ...
2
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1answer
22 views

Extrapolate data points from a series of averages

Given a series of data points: $$\begin{array}{c|c|c|c|c|c|c|c|} & \text{Monday} & \text{Tuesday} & \text{Wednesday} & \text{Thursday} & \text{Friday} & \text{Saturday} ...
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1answer
15 views

Find the marginal density functions for X and Y.

I know that to find the marginal density functions for X and Y, I need to integrate f(x,y) wrt y and x respectively. My problem here is that I don't know what f(x,y) is so can anyone give me some ...
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43 views

Minimum-variance and minimum-divergence estimator

Given a parametric family of distributions $\{P_\theta \colon \theta \in \Theta\}$ and a sample $X \sim P_\theta$, an estimator $T^\star(X)$ of the parameter $\theta$ is said to be a uniformly ...
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1answer
68 views

When to we accept a hypothesis when using Wald test statistic? [closed]

Hello I had to test two hypothesis, one hypothesis gave a wald test statistic with value 0.00015 and the other a value of approximately 40. Is it true when I then say that we accept the hypothesis ...
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1answer
78 views

How many combinations do I need to create to sample 90% of possbile combinations?

I am doing a molecular biology experiment. I have a set of 24 DNA sequences, and I am randomly pairing them together. So, there should be, I think 276 possible combinations (sum series from 1 to 23). ...
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22 views

How to find the expected value of a random variable when it is observed only in a range?

I want to estimate an average livespan of a user account which is a time-difference in days between the account open and account closed. I observe the livespan of accounts that were closed before the ...
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17 views

Significance of variance in small samples

Say I am a school teacher of a class of 18 students. At the end of the year, my headteachers says my final results are below the national average. How can I test whether the difference is ...
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13 views

Sum of the $L$ maximum elements from a collection of RVs

Given a collection of $N$ $\chi^2$-distributed RVs (iid, 2 degrees of freedom), $\{X_1,\dots,X_N\}$ what is the distribution of the sum of the $L$ largest of them? That is, what is the distribution ...
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119 views

UMVUE for $\theta^2$

Let $X_1,...X_n$ be a random sample with distribution $\text{Normal}(\theta,1)$. Find the UMVUE for $\theta^2$ What I´ve done so far: I have already shown that $T=\sum_{i=1}^nX_i$ is a complete ...
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1answer
77 views

2 simple statistics questions regarding probability and means.

Fire alarms go off in the engineering building an average of 13 times per year. Find the probability of more than one fire alarm going off in the month of December. For this one, I am uncertain on ...
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How to computing Wald test in R based on what I have so far?

I have to simulate an AR(1) process with $\rho = 0.5$ and then estimate $\rho$ based on the first 100 $X_T$ values I did this the following way: ...