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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Expectation and variance of occurrences of patterns in a random text

I'm not sure how to complete the following problem from the book An Introduction to Bioinformatics Algorithms: Problem 9.1 Derive the expectation and the variance of the number of occurrences ...
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27 views

Convert list of test results into a comparative score

My students have completed a touch typing test that have given them a WPM (words per minute) result. I now want to give a score to those results out of an arbitrary number, say 10. I will later use ...
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135 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
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67 views

What is the fastest method for factoring this

I want the square root of $.09e^{-2t} + .24e^{-t} +.34 +.24e^t + .09e^{2t}$ I need this to find the moment generating function given it's product with itself (the product of two identical moment ...
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79 views

What does this notation mean? $f(x|\theta)=\frac{3x^2}{\theta^3}I_{(0,\theta)(x)}$

Particularly I want to know what the meaning of $I_{(0,\theta)(x)}$ is here.
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24 views

How is a Brownian motion a collection of random variables

In financial mathematics textbooks I have seen it described as a collection of random variables (in the statistics sense of the term random variable), but is that just because at any given moment, ...
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129 views

Gaussian with a linear combination random variable mean

A very simple (looks like...) statistical problem, however I don't even know how to name it in a formal way... Suppose in a Bayesian framework I have random variables $y, x_1,$ and $x_2$, $$f(x) = ...
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64 views

An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
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45 views

Expected length of a random human being is the same as average of the complete human population

Let $n$ be the number of people in the world. And let $X_i$ be the corresponding length in centimeter. I'm trying to prove that $1/n \sum_{i=1}^n X_i=EX_1$. \begin{align} EX_1 &= \sum_{x=0}^{251} ...
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96 views

Which confidence interval is correct?

I have time series predictions with 50% and 90% confidence intervals. 50% of observations lie below the 50% confidence interval (IIUC, this should happen only 25% of the time), but all observations ...
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32 views

Comparison of variances in small samples

I'm designing an experiment whereby we are attempting to have participants perform data analysis, where the participants estimate a set of values from some real data. The problem is we have two ...
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25 views

Calculation of ${\rm E}[e^X]$ where $X$ follows a uniform distribution over $(1,2)$

To compute ${\rm E}[e^X]$, in which $X$ has uniform $U(1,2)$ distribution. Here $f(x)=1$ for $x\in (1,2)$. The formula is $\int_0^1 e^x \,\mathrm dx$. Is the answer $e^2 - e^1$?
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41 views

Complement of A or B

I have a small general question.. Let's say we have two events $A$ and $B$. Is the probability that $A$ or $B$ will happen, the complement of the event that the complement of $A$ and the complement ...
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33 views

For a set of average scores gathered from a varying number of individual scores, what formula is needed to compute a weighted rating?

I'm currently running a competition of sorts in a group I'm in where we play some video games and log the scores for those games, then determine overall rankings according to the sets of scores each ...
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1answer
66 views

Understanding Poisson Distribution Question?

I have the following question: I have the following formula: However, it is unclear to me how to extract what I need. Would U be equal to 17 for an average of 17 per minute and X be equal to 25? ...
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66 views

If the prior distribution of $\lambda$ is a gamma distribution with mean 1.4 and std dev of .5 find the appropriate values and $\alpha$ and $\beta$

If the prior distribution of $\lambda$ is a gamma distribution with mean of 1.4 and standard deviation of .5 with the form $\pi(\lambda | \alpha, \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} ...
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17 views

Average of several harmonic means

I would like to know if it is possible to calculate the mean and the standard deviation for several harmonic means. More specifically, I have 20 harmonic means. And every harmonic mean was calculated ...
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218 views

Find Bayes estimator of $\theta$

I've got this exercise, which I'm trying to work off using an example, but the example seems very different so I'm not sure if what I'm really doing. I've got a loss distribution for $\theta$: ...
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107 views

expectation of norm of orthogonal projector

The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace $T := span\{U1, U2\}$ where $U1$ is a random vector uniformly distributed over unit ...
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60 views

Find marginal distribution for Pareto prior

I have the following problem: The prior distribution for $\theta$ is distributed $\pi(\theta) = \frac{aP^a}{\theta^{a+1}}$, $\theta >P$ The likelihood for X is uniformly distributed, i.e. ...
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54 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
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18 views

Help with comparing risk function

QUESTION (TPE, Lehmann & Casella, 1.22) Let $X$ take on the values 1 and 0 with probability $p$ and $q$, respectively, and assume that $1/4<p<3/4$. Consider the problem of estimating $p$ ...
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34 views

Probability Distribution in this situation

Good Day, I have a computer science question i need to solve. I am looking to get a probability of events that occur on a daily pattern. For example: A week would look as such ...
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108 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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79 views

What is the probability that a customer waits for lesser than 3 minutes?

The rate of service is exponential and the service rate is 12 customers served per hour. The arrival of customers is in a Poisson distribution at the rate of 30 per hour. There are 3 servers and the ...
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36 views

Predict amount of goods to order?

Someone familiar with prediction problems, where you need to predict how much goods to order given some history? For example, this week you have the history of how much apples you ordered, how much ...
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1answer
94 views

Random variable and Poisson distrubtion

Given that $X$ is a random variable having a Poisson distribution, compute the following: (a)When $μ=0.5$, $P(X>3)$, My attempt: $1-e^{-.5}-e^{-.5}-e^{-.5}\cdot0.5-\frac{e^{-.5}\cdot0.5^3}{3}$ ...
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264 views

Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility: I have monthly index values starting in Jan 2005 ...
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1answer
102 views

Find joint probability P(X=0, Y=0)

I have this problem where I'm not too sure on how to proceed. I need to calculate $Pr(X=0 $ and $ Y=0)$ using the following information: The conditional distributions $f(x|\theta)$ and ...
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1answer
259 views

Maximum likelihood estimator of minimum function with exponential RV and a random number

I'm having some problems with the following assignment: Let $X_1, X_2, ...,, X_n$ be samples from a exponential distribution with parameter $\lambda$, and let $c_1, c_2, ..., c_n$ be a sequence of ...
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75 views

Tail bound sum of unbounded RVs of given mean and variance?

Let $X_1,\dots,X_n$ be independent variables, $X_i$ having mean $\mu_i$ and variance $\sigma_i^2$. Let their sum $S = \sum_{i=1}^n X_i$. Of course, $S$ has mean $\lambda = \sum_{i=1}^n \mu_i$ and ...
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31 views

Pyschology Statistics help in understanding a formula

I don't quite understand why we use this formula for sampling distributions between two proportions. I have an upcoming midterm in my psych class and I feel that none of this stuff is sinking in. My ...
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1answer
23 views

Question about order statistic

Let $X_1$, $X_2$ be two samples draw from a continuous distribution, then I think there is no reason to say that $X_1\leq X_2$ or $X_1\geq X_2$, so we may have $$P[X_1\leq ...
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142 views

Probability density function of a summation of continuous random variables

Let $Z_{i} = \tau + X_{i}$, where $X_{i}$ is a exponential random variable ($X_i \sim \varepsilon(\lambda)$), $0<\tau, \lambda < \infty$ Assume $X_i$ are independent random variables. Suppose ...
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83 views

Expected value of a certain game

Consider a game with n players playing m rounds. Players play in the same time. Player wins k dollars with probability p and loses everything otherwise. If player loses the rest divides his money ...
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36 views

Statistics Normal probility

A chromosome mutation believed to be linked with color blindness is known to occur, on the average, once in every 10,000 births. If 200,000 babies are born this year in a certain city, use Normal ...
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1answer
24 views

Poisson Process Basic Question

The sum of independent interarrival times for the poisson process is a gamma random variable. in general does the sum of exponentials have to be independent to sum to gamma? also this would produce ...
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82 views

Calculate the confidence interval for a single variance given two variances (using pooled variance)

I am having trouble trying to interpret why I'm given some value for a homework exercise. It goes like this: Suppose we're studying the time it takes for a certain industrial process to complete. ...
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49 views

Number of rounds to find out the winner in nondeterministic game

Imagine, I'm organizing competition for AI Yahtzee players (or some other game, poker, backgammon etc.). I want to find out who plays better: player A or player B. If they play just one game, winner ...
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35 views

Expected value of $x^TR_x^{-1}x$

Assume that the inverse of the correlation matrix of the n dimensional column vector $x$ exists and equals $R_x^{-1}$, show that $E\{x^TR_x^{-1}x\}=n$.
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25 views

Result depending on a lot of variable where a change won't really matter much, any principle to describe this?

I have a value $A =f(p_1,p_2,p_3,...,p_N)<0$ with certain values of $\bar p$. The function $f$ is a multilinear function. Now the value of $A$ is very rarely above $0$ with a lot of different ...
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48 views

Probability of Combinations which have different probabilties.

I cannot figure this out. The problem goes A boy has a bag filled with balls: 1 red 2 green 4 Blue 7 White A child plays a game in which he withdraws one ball, writes down its color, and then ...
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202 views

Derive probability density function from moment generating function

I am given this information: $\tau$ is a constant, $\infty > \tau >0$ Let $X \sim G(n, \lambda)$, where $n \in \{1,2,...,\}$ ($G$ is referring to gamma distribution) Let $Y = n \tau + X$ ...
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49 views

Can infinite random sequences be asymptotically compressed?

A number $0.5<p<1$ is chosen at random and given to two people A and B whom are allowed to communicate before beeing separated. A is then given a sequence S of N random bits where each bit has ...
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236 views

Mean/ Expected Value of $X^4$

$\def\Var{\mathop{\rm Var}}$Can anyone help me prove that Expected Value of $X^4$ is $3\Var(X)^4$, if the Expected Value of $X$ is zero and $\Var(X)$ is the Variance of $X$? Thanks!
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179 views

Calculating the expected profit with Probability A level maths CIE

Company sets up display of 20 fireworks! for each firework, the probability that it fails is 0.05,independently of other fireworks the probability that more than 1 firework fails is 0.264 the 20 ...
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92 views

Prove random vector with covariance matrix $\Sigma$ has non-degenerate distribution iff $\Sigma$ is positive definite

Let $X$ denote a $d$-dimensional random vector with covariance matrix $\Sigma$ satisfying $|\Sigma| < \infty$. Prove $X$ has non-degenerate distribution iff $\Sigma$ is positive definite. ...
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2k views

How to calculate the probability of a normal distribution with unknown mean and unknown variance?

How do you calculate the probability of a normal distribution with unknown mean and unknown variance? If a problem stated, for example, that 15% of the time sales are more than 15,000 and 20% of the ...
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74 views

Derive asymptotic behavior of inverse of the normal cdf with respect to 2^n

I have a normal distribution $\mu = 0$ and $\sigma = 0.58n$ where $n > 0 $ and I am trying to derive the asymptotic behavior of the following equation: ...
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70 views

What does mean this notation after the P?

It looks like small capital letter lambda, Λ. What is its purpose?