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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Prove that $\mathrm{E}[X\mid A] = \mathrm{E}[X]$ for an event $A$ independent of random variable $X$

I am a novice at statistics. I have approached the problem in the following way: $$\mathrm{E}[X\mid A] = (x_1\mid A \cdot P[x1\mid A] + x_2\mid A \cdot P[x_2\mid A] + \ldots + x_k\mid A \cdot ...
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1k views

Four married couples, eight seats. Probability that husband sits next to his wife?

There are four married couples and eight seats. When they sit, what is the probability that husband sits adjacent to his wife? The answer to this problem is 12/35 I can arrive at this answer when I ...
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1answer
157 views

Completeness and MLE of a discrete distribution

The random variable X takes on the values 0, 1, or 2 according to the following probability distribution: $P(X = 0) = P(X = 1) = p^2, \ and P(X = 2) = 1 - 2p^2, for \ p \in C.$ (1) Determine whether ...
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1answer
426 views

CDF and conditional probability

The cdf of $T$ is given by $F_T(t)=1-e^{-\lambda t}$ for $x>0$. Show that $P(T>t+s\mid T>s)=P(T>t)$. Find the MGF of $T$. I don't really know what's going on. Do I substitute $t+s$ into ...
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2answers
1k views

What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid. It is possible to relax these assumptions. For example relaxing the independent distribution ...
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1answer
231 views

What information can we obtain from a uniform, random variable?

Suppose that the time students wait for a bus can be described by a uniform random variable X, where X is between 0 minutes and 60 minutes. a) sketch the density function b) what is the probability ...
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1answer
46 views

Scalar generalization of variance for multi-dimensionnal random variables

I am wondering how to define a scalar generalization of variance for multi-dimensionnal random variables. In the case of simple multi-dimensionnal spaces where all dimensions have a similar behaviour, ...
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1answer
77 views

Simple Linear Regression

$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|} \text{Obs}\# & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\\hline X & 5 & 8 & 10 & 4 & 5 & 12 & ...
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1answer
53 views

Stochastic Processes Question

Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
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1answer
300 views

Does “50/50 chance of.. . ” convey information?

I distinctly remember the professor in the undergrad introductory systems & control course saying that "when weather forecasters say there's a 50% chance of precipitation, they are conveying no ...
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1answer
140 views

Probability with Chi-Square distribution

What is the difference, when calculating probabilities of Chi-Square distributions, between $<$ and $\leq$ or $>$ and $\geq$. For example, say you are asked to find P$(\chi_{5}^{2} \leq ...
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1answer
93 views

Clasification of parameter estimation method

Consider that $P$ is the water pressure coming out from a valve A, therefore, the population is all the valve A pressure values. Let $P_{dif}$ be defined as the difference between the maximum and the ...
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1answer
35 views

Is this proposition correct?

$P(A\mid B)\leq \frac{a+b-1}{b}$ where $P(A)=a,P(B)=b$. I found this example problem on notes I took in class but I forgot to copy down how the prof proved the proposition. Maybe I miscopied the ...
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1answer
554 views

How do you find the mean and variance with new observations?

When given a mean and variance of a sample, without knowing the observations, how would you then find the new mean and variance given more observations? Any help with this would be much appreciated. ...
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1answer
40 views

Probability of all parts functioning

For a trip to be successfully launched, 100 different parts on the ship must all be functioning properly. The probability for each parts failing, $p$, is $0.0001$. i. What is the probability that the ...
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1answer
1k views

Finding the Maximum Likelihood Estimator of a Median

Let $X1, \dots, Xn$ be a random sample of size n from the continuous distribution with pdf $f_X(x|\theta) = \frac{e^{-x}}{1-e^{-\theta}} I(x)_{[0,\theta]} I(\theta)_{(0, \infty)}$. (1) Find the ...
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1answer
527 views

probability distribution $X, Y$ and $X+Y$

A box contains $5$ ticket, $\{ 0 , 0 , 0 , 4 , 4\}$. Drawing two tickets at random w/o replacement. $X$ be the sum of the first two draws and Y be the outcome of the first draw. Question: Find ...
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2answers
2k views

Find the variance of $(\bar X^2)$ without using Moment Generating Function

Is there an easier way to find the variance of sample average squared $(\bar X^2)$ without using the moment generating function? $X\sim N(\mu, \sigma^2)$ I know that $Var(\bar X^2)= E(\bar X^4) ...
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1answer
56 views

minimum of absolute value

If we consider the following problem $$ \mathbb{E}[(Y-y)^2 | X=x] $$ I can easily show that the minimum with respect to $y$ occurs at $$ y=\mathbb{E}[Y |X=x] $$ How can I find the minimum of $$ ...
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1answer
271 views

Multivariate Gaussian equivalent for a Gaussian integration identity.

For a one-dimensional x, $$\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}dx=\frac{1}{2}\int_{-\infty}^{\infty}e^{-x^{2}}dx$$ This can be shown through integration by parts. There is a good derivation of ...
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2answers
610 views

What is the probability that two samples represent the same normal distribution?

Yes, it's a basic question. But, I have searched about 25 web pages for this and found only things that were irrelevant or incomprehensible. So I have indeed tried. My question is: I have two ...
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1answer
245 views

Normal distribution theoretical moments

how we can show that the following equality holds $E[(x-\mu)/\sigma]=0$ $E[(x-\mu)^2/\sigma^2-1]=0$ $E[(x-\mu)^3/\sigma^3]=0$ $E[(x-\mu)^4/\sigma^4-3]=0$
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1answer
314 views

average waiting time

who can help me to resolution of this statistic exercise? below the track: Caio go in a bank,the number of customers ahead him are described by a Poisson random variable of parameter a>0. Calculate ...
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25 views

How to a sample of a time series of zeros and ones with given autocorrelation function?

Let's say, I have an autocorrelation function: $$C(\text{lag}) = \text{lag}^{-\gamma}$$ How can I generate a sample of a time series of, say, $0$'s and $1$'s, such that their sample autocorrelation ...
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1answer
142 views

What is Kendall Tau's co-efficient and Pearson co-efficient

I came across these two terms in a paper about Natural Language Processing. So I looked both of them up on the net and couldn't understand a thing. So far I think their a method of comparing two ...
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1answer
351 views

Percentage point of Normal Distribution.

Let $$X \sim N(65,64) $$ Find the lower $2$% point for $X$; that is, find the value of $x$ such that $Pr(X<x) = 0.02 $ i know i need to do something like $\frac{X- 65 }8 = ... $ but not ...
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1answer
138 views

“Practice Exam” probabilities

A practice exam with 6 questions is prepared. 3 of which will be very similar to ones on the actual exam. a) How many possible ways can the exam be written? (i.e. how big is the sample space of ...
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1answer
48 views

log likelihood and log error

In statistics one often hears the phrases "log likelihood" and "log error". Why is it natural to consider the logarithm of these guys? My wild guess is that the Gaussian is a pretty "normal" ...
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0answers
231 views

Statistics Question - Normal Distribution

The scores of a final exam have a Normal Distribution with mean $75$ and standard deviation 6. An independent sample size of $9$ is drawn from this distribution. The corresponding random variables are ...
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70 views

What does the '•' mean in $(a(x,y, t) \cdot b(y,t))$

I'm working on statistics and have seen an equation that has a section like that shown above. $Max(_{y}\epsilon_{nbh(x)}) (a(x,y, t) \cdot b(y,t))$. I am unsure what the '•' means so the whole ...
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466 views

How to calculate the highest/smallest possible value of the variance of two random variables mean?

Two random variables $X$ and $Y$ have a common expected value $E(s)$ and a common variance $Var(s)$. What's the highest possible value of the variance of their mean, $var ((x+y)/2)$? What's the ...
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1answer
436 views

Weighted Standard Deviation for Histogram Bin Height

I'm plotting some binned data in the form of a histogram. Say I have 10 data points, each composed of a bin to be placed in, and then a "height". Then I might have something like: Bin Height 0 - ...
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1answer
508 views

Statistics Question: Revenue & Cost

Data for costs of production for a firm every month from January 2001 to December 2002. Data is denoted by $d_1, d_2,...,d_{24}$. The following info is calculated from the data set. Mean cost = ...
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1answer
206 views

Finding the efficiency of an unbiased estimator

I have a random sample drawn from a $N(\theta,\sigma^2)$ distribution with $\sigma^2$ known. I am trying to estimate $\theta$. I need to calculate the efficiency of the unbiased estimator, ...
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2answers
115 views

Construct a Confidence Interval of $95\%$

Based on a random sample of 20 values from a normal distribution with mean $\mu$ and variance $\sigma^{2}$, it was calculated that $\bar{X}=8$ and $s=4$. Provide a $95\%$ confidence interval for the ...
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79 views

Does this count as a Monte Carlo simulation?

Let's say I have a group of robots that walk on a 11x11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different ...
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1answer
247 views

Game Theory: determining the value of the foxhole game

"A soldier can hide in one of five foxholes, and a gunner can hide in four spots: A, B C, and D. The configuration looks like this: 1 (A) 2 (B) 3 (C) 4 (D) 5. If a shot is fired at a location and the ...
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473 views

On the empirical mean and variance of a Poisson i.i.d. sample

Let $X_1, X_2, \ldots, X_n$ be a random sample from a Poisson($\lambda$) distribution. Let ($\bar{X}$) be their sample mean and $s^2$ their sample variance. Show that ...
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1answer
126 views

How can I solve this integral?

How can I solve the following integral? $$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...
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1answer
491 views

Conditional Probability Problem With Non-Disjoint Events

The problem is: A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let Bbe the event that the ...
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1k views

Conditional Probability Problem With Boxes

The problem I am working on is: One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the first box ...
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1answer
58 views

Gambler tosses coin PGF

A gambler tosses a coin repeatedly until it comes up tails. He gets £1 for each head that comes up before that happens, and pays back £1 for the tail. Find the probability generating function for his ...
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1answer
663 views

Probability of 5 Card Poker Hand with Only Two Suits (Method)

Probability of 5 Card Poker Hand with Only Two Suits: Pick a card from any suit: 52 Pick a card from a different suit: 39 Pick three more cards from either of the first two suits: $24*23*22$ So, ...
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549 views

Intuition Of Conditional Probability Equation

I was wondering if any one of you had any intuitive insight regarding the conditional probability equation, $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$. In my textbook, they give a mere definition, ...
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1answer
530 views

How to count $n$th percentile from normally distributed random variable?

I have normally distributed random variable $X\sim \mathcal N(100,225)$. How to count $n$th percentile? In my case I need lower quartile - $x(0.25)$.
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3k views

Combinatorics and Probability Problem Concerning Poker Hands

The problem I am currently working on is: In five-card poker, a straight consists of five cards with adja-cent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king ...
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1answer
199 views

Given the sample that not Independently and Identically Distributed

So I'm working on a question that wants me to consider the sample mean $\mu_1$ (estimator of $\mu$) when $n=3$ to be $μ_1=X_1+X_2+X_3$. Now first I assume that they are independently and identically ...
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1answer
7k views

How to check if my dataset is normally distributed?

I have data sets (measurements) and I need to know if values are normally distributed. I would like to get this information programmatically in my application and not via plotting and checking it ...
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1answer
18 views

Clarification on a basic stat problem

I need a clarification on this homework problem I'm getting done. The question is as follows: For each positive integer $n$, let $P_n = \Big (\frac 12 \Big)^n.$ Consider the events $A = \{n:1 \leq ...
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103 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...