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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The CDF of an i.i.d. sample realizations

Is it possible to draw the CDF of the empirical measure $\hat{P}_x$ for a i.i.d. sample realizations where $X_1 = 0.3$,$X_2 = 5$,$X_3 = 1.5$, $X_4 = 3.4$ without knowing the distribution?
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83 views

Calculate probability that random integer already exists in set of N random integers

I have a set of N random integers between A and B. Assuming that my random number generator is equally likely to return any integer between A and B, how can I calculate the probability that the next ...
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1answer
90 views

Strong Law of Large Numbers Under a Transformation

I have some random variable, $x$, distributed according to a probability density function (pdf), $f\left(x\right)$. The Strong Law of Large Numbers (SLLN) implies that, for an expected value, given ...
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64 views

How do I find the mean with this exponential?

p(y) = c/y^4 I need to find the "mean & variance" of this exponential density function. Some pointers or thoughts that would explain would be most helpful. ...
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4k views

Calculating missing data points from standard deviation and mean

I have to figure out what to missing data points are from a set of 10 scores. The mean of the 10 observed scores is 20.0 and the standard deviation is 6.0. The observed scores are listed below, with ...
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483 views

How many degrees of freedom?

I have another question on my stats homework, relating to degrees of freedom. However, I have NO idea how this question even relates to degrees of freedom! I understand degrees of freedom on their ...
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190 views

How do I account for opposite answers when calculating city block distances?

For a stats class, I have a question in which an occupational therapist uses a checklist about meal preparation. The checklist consisted of five statements to which persons responded using the ...
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568 views

PMF of a geometric random variable and binomial random variable

We have been discussing pmf. Can someone please show me A. That the pmf for a binomial random variable sums to 1 B. That the pmf for a geometric random variable sum to 1 So I may see the difference ...
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240 views

Can you please help me find the Moment generating function of the negative binomial when it looks a bit different?

f(x) = $\binom{(r+x)-1}{x-1}p^{r}(1-p)^{x}$ noting 0 < p<1 and r>0 $$\sum_{X}^\infty \binom{(r+x)-1}{r-1}p^{r}(1-p)^{x} =1.$$ x=0,1,2,3......... I think it looks like this: $$\sum_{X}^\infty ...
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143 views

How to find the mean [E(x)] of this function. Please i know i can find variance after this is shown

Then how do i find the mean of $$\int_0^\infty\frac4{\sqrt{\pi} b^3}x^2e^{-x^2/b^2}dx;$$ if so do how do i do this using the gamma distribution? is there a short cut or do i have to do integration by ...
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86 views

Show that an estimator has a lower variance than another estimator

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sufficient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
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Showing that statistic is unbiased

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
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1k views

Covariance Matrix in Weighted Least Square Estimation

I am new to linear algebra and I have the following doubts: In weighted least square estimation of the system $Ax = b$ we minimize the weighted value of the error $e = b - Ax$ and the best $\hat{x}$ ...
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237 views

Baseball Probability question

Consider. You throw 3 baseballs at a target. Each baseball has the probability (independent) p= 3/5 of a hit. What's the probability that all three hit? If it is such that a target will be knocked ...
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400 views

Integration by parts and polar coordinates

Can someone please show me how to integrate $$\int_0^\infty\frac4{\pi b^2}x^2e^{-x^2/b^2}dx\;?$$ please show steps how to integrate this problem. This is what i have so far. $$\frac4{\pi ...
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45 views

Showing that a set of data with a normal distibution has two distinct groups when you know which point is in which group vs when you don't

Assuming you have a set of data containing two groups, assuming the data would be gaussian distributed, for example height of people, knowing that there are men and women, you can show that the groups ...
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689 views

monotone functions and pdf's

Could you please show me step by step? Also how does the probability integral transformation come into play? "If the random variable $X$ has pdf $$ f(x)= \begin{cases} \tfrac{1}{2}(x-1)\quad \text{if ...
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860 views

Least squares estimator of mu

The question is: Assuming that $y_i = \mu + \epsilon_i $,$i = 1,\ldots,n$ with independent and identically distributed errors $\epsilon_i$ such that $E[\epsilon_i] = 0$ and $Var[\epsilon_i] = ...
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192 views

Straight line through data by eye - least squares? [closed]

I heard an interesting fact a while ago about how people draw a line through a cloud of points on a scatter plot. Usually, when calculating lines of best fit, we use the minimal the sum of squares of ...
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1answer
47 views

Average runtime based on X failures

I have a power generator that runs until it fails x times and stops. I want to try to predict how long it will run on average based on known probablities for failures during its operation. The ...
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124 views

Understanding importance sampling

From Wikipedia Let $X:\Omega\to \mathbb{R}$ be a random variable in some probability space $(\Omega,\mathcal{F},P)$. The basic idea of importance sampling is to change the probability $P$ so ...
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51 views

Determining The Best Arrangement Of Skills

I have no idea if this question is correct for this forum, but I figured I'd give it a shot. I am trying to figure out the best order to execute skills in a game (I know). Anyway here is the skill ...
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0answers
36 views

statistics- iteration of the alpha parameter in a beta function

I'd like to know whether starting from a Beta distribution and iterating it in the way described below I get a stationary (beta) distribution again. More specifically this is the problem I am facing: ...
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34k views

What is the equation used to calculate a linear trendline?

In excel it is done automatically but how to manually calculate a linear trendline over a set of points?
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114 views

Blackjack Basic Strategy

I'm reading Beat the Dealer and I'm confused why you hit when holding a hard twelve against a dealer up card of two or three. You stand on a hard twelve if the dealer shows a 4-6. Can someone ...
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440 views

Exponential Distribution MLE

The lifetime of a type of component has an exponential distribution with rate λ per hour. Ten of these components were tested but the only recorded results were that 3 components had failed within 100 ...
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28 views

Average duration based on changing probabilty

Given a game where the player has some probability of dying and this probability increases each second as the game is played to some maximum probability. The player has three lives and then the game ...
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402 views

Determine whether a statistic is sufficient, given the probability density

Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x ...
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538 views

How to Calculate the Conditional Distribution of a Sum of Uniform Random Variables given the Observed Range

Suppose $X_1$ and $X_2$ are taken at random from a uniform distribution on the interval $[\theta - 1/2, \theta + 1/2]$, where $\theta$ is unknown $(-\infty < \theta < \infty)$. Let $Z = Y_2 - ...
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3answers
866 views

Binomial Probability Word Problem

This problem appears very simple, but I am almost positive that it should not be so simple. For the question below I reason that the probability for part a is 0.6 and part b is 0.6 as well. What am I ...
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115 views

Given that $X$ is normal, find the probability that $(X-10)^2 <12$

Suppose that $X$ is a random variable that has a normal distribution with mean = 5 and standard deviation = 10. Evaluate the following probabilities: $\mathrm{Prob}((X-10)^2 < 12)$
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Does changing two events to mutually exclusive, change their probability?

Does changing two events, $A$ and $B$ for example, to mutually exclusive in a new/altered situation, change their probability from the precondition? Would the $P(A)$, probability of $A$, now equal the ...
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862 views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
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240 views

Arrange n black and 4 white marbles

I have this question that I can't seem to wrap my brain around. It goes something like this. How many ways are there to arrange n black and 4 white marbles so that every white marble is adjacent to at ...
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13k views

How to calculate median and standard deviation from histogram?

I have a 256 bin histogram of an 8 bit image. Is it possible to calculate the Median, Variance and Standard deviation of the image from the given histogram? If yes, then what is the procedure? ...
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419 views

Proportion of Values

I'm not quite sure how to do this problem: Calculate the the proportion of values that is 3 in the following data set: 2, 3, 3, 6, 9
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103 views

Probability of character series in random word

I have a word with 13 characters. Each character is random and alphanumerical (a-z, 0-9). So for each character there are 36 possibilities: 10 numbers and 26 letters. I am trying to find out the ...
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230 views

Confusion about covariance

$\newcommand{\Cov}{\operatorname{Cov}}$ $X_i$ from $i=1,\ldots,n$ is a set of random variables The following confuses me: $\Cov\left(\sum_{i=1}^n X_i, \sum_{j=1}^n X_j\right) =$ the sum of all ...
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59 views

Find factor of sample elements given the median

I have a sample $S(x)$ containing $n$ elements: $$S(x)=\{ s_1 x, s_2 x, \ldots, s_n x \},\qquad s_i \in \mathbb{R}, x\in \mathbb{R^{+}}$$ Every element in the sample is multiplied by $x$. Now ...
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178 views

What is the meaning of $P(A \cup B)$ in probability and statistics?

In probability, letters $A$ and $B$ are used to denote various events. Then we write $P(A)$ for the probability of event $A$ happening. Same for $P(B)$. But I often see the notation $P(A\cup B)$ as ...
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159 views

sale price, interest per annum and percentage increase or decrease

Mary bought a computer on hire purchase. If the cash price on the computer was $3 million and 25% was down payment if bought on hire purchase. If the balance was paid in 12 monthly installment at 12% ...
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344 views

I am confused about Bayes' rule in MCMC

Bayes' rule appears to bevery simple at first sight, but when studied deeply I find it is difficult and confusing, especially in MCMC applications when multiple parameters need to be estimated. For ...
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23 views

Smoother Generalized Mixed Linear Models

I wondered if anyone could help me figure out what a smoother was in regard to Generalized Mixed Linear Models or point me in direction of some relevant litterature. Regards!
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608 views

Confusion on the meaning of confidence interval

The standard deviation of test scores on a certain achievement test is 10.9. A random sample of 60 scores on this test had a mean of 72.1. Based on this sample, find a 95% confidence interval for ...
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263 views

Hypothesis Testing Normal Distribution

I am unsure how to approach this problem: Consider a random variable, X. Our hypothesis is that X~N(0,1) or X is standard normal distribution. If we observe that X = 2.2. What is the likelihood of ...
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3k views

Linear transformation of normal distribution

Not sure if "linear transformation" is the correct terminology, but... Let $X$ be a random variable with a normal distribution $f(x)$ with mean $\mu_{X}$ and standard deviation $\sigma_{X}$: $$f(x) = ...
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762 views

$P(B|A)=1-P(\sim B|A)$ are these equivalent?

The probability of $B$ given $A$ and one minus the probablity of $\sim B$ given $A$. So, $$P(B|A)=1-P(\sim B|A)$$ The events are not necessarily mutually exclusive, I just wanted to know if the ...
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384 views

Calculating critical values in chi distribution

I am learning about chi tests and I've understood everything so far except how to calculate critical values. I don't want to take number of degrees of freedom and probability of exceeding the critical ...
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582 views

Binary Logistic Regression Model Processing

Thanks for showing interest and wanting to help out. My aim is to develop a model that - as accurately as possible - predicts how entities in a population will either cooperate or defect, as a % of ...
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2k views

Distribution of $-\log X$ if $X$ is uniform.

For $X$ and $Y$ random variables; $X$ follows the uniform distribution. (1): if $Y=-\log X$ (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}. Why ...