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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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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224 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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44 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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550 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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943 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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136 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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587 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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284 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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28 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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203 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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94 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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51 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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41 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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1answer
172 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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1answer
57 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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44 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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54 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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315 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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58 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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2k 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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544 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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162 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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6k 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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121 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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100 views

What does mean this notation after the P?

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

Calculating an average on normal distribution

Given the fair dice, if the result is $1$ or $2$ the profit is $3$USD, if the result is $6$ you don't win or lose anything, for every other result you lose $2$USD. What is the average profit, that ...
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43 views

A problem on distribution functions

I have a quick question here. From the definition of a distribution function (DF), $\text{A real-valued, nondecreasing, right continuous function} \; F \; \text{defined on} ...
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43 views

In statistics, what is the meaning of $Z_{0.3}$

What is the meaning of $Z_{0.3}$ and how do I calculate it? I know it was calculated this way: $$Z_{0.3} = -Z_{0.7} = -0.52$$ I tried to follow the General Distribution table but I can't seem to ...
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75 views

Confused about notation: difference between $\prod_{i=1}^np(x_i)$ and $\prod_{i=1}^np(x)$

In my information theory book by Cover and Thomas, at the beginning of the channel coding theorem, it's written: "Each entry in this matrix" (the matrix of the randomly generated code) "is ...
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1answer
30 views

What does test the model for significance mean?

Say that I have a model $\hat Y_i=\hat \beta_0+\hat \beta_1X_{i1}+\hat \beta_2X_{i2}+u_i$, where $X_{i1}$ is a binary variable that takes the value 1 if female and 0 if male, and $\hat Y$ is the ...
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112 views

Find transformation function when densities are known

I need some help with the following probability/statistics problem: Let X be a continuous random variable with density $f_{X}(x) = \begin{cases} \mathrm{e}^{-x^{2}} & \text{}x\text{ > 0} \\ 0 ...
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277 views

Question from “An introduction to measure theory” by Terence Tao [duplicate]

If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} < \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is ...
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2k views

Relation between standard deviation and mean in random processes

In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of ...
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998 views

Normal distribution, chi-square distribution and t distribution combiened

How to prove that when X is from Normal Distribution and Y is from Chi-square Distribution with parameter f and X,Y are independent then X/sqrt(Y/f) is from t distribution with parameter t? I got ...
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357 views

Prove $E(|X-\mu|) = \sqrt{\frac{2}{\pi}}\sigma$

Prove $$E(|X-\mu|) = \sqrt{\frac{2}{\pi}}\sigma,$$ if $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. I don't know where to begin with this problem and would like help! ...
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104 views

How many items to acquire to get a full collection? (random chance involved)

The problem: My daughter wants to have a collection of certain Kinder Surprise toys. There're 10 different toys in this series. Assuming the chances of getting each toy are equal, how many Kinder ...
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1answer
1k views

Predicting the number of orders from future customers

Tamara is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 450 customers ordered turkey, 375 customers ordered ham and 250 customers ...
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652 views

Determine if being on honor roll gives an advantage in being assigned to the math class

Andrew believes the honor roll students at his school have an unfair advantage in being assigned to the math class they request. He asked 500 students at his local school "Did you get the math ...
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48 views

conditional probability that randomly chosen

In a certain village sports club, 46 % of members play football, 36 % of members play cricket, and 17 % of members play both games. What is the probability (between 0 and 1) that a randomly chosen ...
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1answer
5k views

Margin of error and $98\%$ confidence interval question for stats people :)

The question is (this is homework, for an online class, no teacher so at times confusing) Carl conducted an experiment to determine if there is a difference in mean body temperature between men and ...
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1answer
47 views

3 dices probability that number 5 comes up on exactly two of the three dice?

I roll three different loaded dice. For the first die the probability of getting a 5 is 0.7, for the second die the probability of getting a 5 is 0.48, and for the third die the probability of getting ...
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204 views

Proportional random sampling from a set of variable combinations

I have this table of variables, they can have the following values: $v_1 \in \{1,2\}$, $v_2 \in \{1,2,3\}$, $v_3 \in \{1,2,3,4\}$, I need to generate all the possible combinations $v_1v_2v_3$ of ...
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2answers
27 views

$Z_1$, $Z_2$, and $Z_3$ are independent standard normal variables. Obtain the correlation matrix

Define $X_1= 2Z_1 - Z_2 +3Z_3$ $X_2=Z_1 + Z_2 - Z_3$ $X_3 = 3Z_1 + 2Z_2 + Z_3$ So would the correlation matrix of $(X_1, X_2, X_3)$ just be the coefficients of each variable?
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Minimum number of data points for RMS

In calculating the rms of a group of data points, what is the smallest number of points needed to get a meaningful rms value? Specifically I'm looking at audio files and using an rms value for a small ...
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52 views

What is the distribution of $\frac{1}{2}[(X_1 - X_2)^2 + (Z_1 - Z_2)^2 + (Z_1 + Z_2)^2]$?

Let $Z_1$ and $Z_2$ be a random sample from normal distribution $\mathcal{N}(0, 1)$ and $X_1$ and $X_2$ be a random sample from normal distribution $\mathcal{N}(1, 1)$. Suppose the $Z_i$'s are ...
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153 views

Kalman filtering correlated measurements

I would like to run a Kalman filter over a set of measurements which may (will) be correlated. Essentially, each new measurement contains (say) 90% of the same information from the previous ...
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185 views

probability of exactly, atleast, and expected number

The people discovered that 54% of refugees who ask for asylum in the New York immigration court win asylum, but only 12% are granted asylum in the Florida immigration court. Assume that you randomly ...
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2answers
46 views

normal distribution question?

I was wondering if anyone could explain how to use the mean and standard deviation of a normal distribution to determine the top 5% of population? -It is for homework, but I wont post the actual ...
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143 views

Rearranging logarithmic equation

I've tried hard to rearrange the following equation to calculate the (AGE) ...
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184 views

Why doesn't eigenvlaue based PCA and SVD based PCA produce answers that are exactly the same?

I have been testing the princomp() function in MATLAB: http://stackoverflow.com/questions/19665401/pca-returning-error-undefined-function-or-method-pca-for-input-arguments#19666296 As part of the ...