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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Problems getting the covariance matrix of the ressiduals

In order to get the variance-covariance matrix of the residuals of a linear regression model, I do the following: Considering that the residual vector $e$ is: $e = Y - \hat{y} = XB+\epsilon - Xb$ ...
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20 views

Comparing an isotonic model to an additive model

Say I have a dataset in $x,y$, and say I fit a few different models to the dataset. Examples could be an isotonic regression, a smoothing spline and a simple linear regression. What are some ways I ...
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33 views

How to quantify the effectiveness of a baseball statistics prediction model?

I'm working on a model to predict baseball statistics based on the previous season's statistics. I ran the 2010 stats through my model, giving me predicted values for 2011 for a particular statistic. ...
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1answer
50 views

If $X_n$ sequence of random variables , equally distributed $EX_n=a, n=1,2,3…$ then $\frac{1}{n}\sum_{k=1}^{n}X_k\to^{P}a$

If $X_n$ sequence of random variables , equally distributed $EX_n=a, n=1,2,3...$ then $\frac{1}{n}\sum_{k=1}^{n}X_k\to^{P}a$ (convergence in probability) Proof: (using the fact that convergence in ...
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1answer
36 views

Bayes Theorem hypothetical situation

Let's say I have a friend named Dave. There was a murder committed next door. Dave is most likely the killer. P(dave committed murder)=0.99 However, the probability that Dave would leave a blond ...
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30 views

What are the odds of players drawing a specific card in a round?

FIRST: Thanks for the advice on finding out how to ask my question. I revised my question, and here it is. I hope what I'm looking for is clearer. I'm making a game and I need help figuring out the ...
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1answer
33 views

Probability Q of one event happening but not the second or third…

I'm in a college Prob/stats course and I need help with how to find the answer to this: Jeanie is a bit forgetful, if she doesn't make a to do list the probability that she forgets something is 0.1. ...
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1answer
75 views

Geometric Distribution within a Normal Distribution

So here's the problem: On Interstate 40 north of Chapel Hill, the distribution of speeds of cars is approximately normal with mean 60.6 miles/hour with standard deviation equal to 4.78 miles/hour. We ...
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16 views

Finding Asymptotic Variance

Can anyone help walk me through finding the asymptotic variance for some random variable? For example, let us consider $\sqrt{n}(\bar{y}-\mu) \equiv *$ I think this converges in distribution to ...
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2answers
70 views

Finding variance of number of distinct values chosen from a set with replacement

There are k unique values, and we choose n of them with replacement. Let x be the number of unique values chosen, independently and uniformly. I found the expected value: $$\begin{align}E[X] &= ...
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16 views

How to do hypothesis testing in discrete same sample case

Please have a look at the question in the link(http://postimg.org/image/5h7i2jfi5/). ( I do not have enough 'reputation' to post images here) How can the question be solved? In the question we have ...
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1answer
27 views

How to profile people using clustering

I have a data of customers and i want to split them to segment (profiling). The columns of the data are Amount-Spending, Amount-Bonus, Age, Churn-or-Not. So i clustered my data with k-means. to 3 ...
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31 views

Definition of a statistical test

I'm trying to understand this definition of the notion of a statistical hypothesis test I ran into in a book. However, for some reason, I'm having trouble deciphering their given explanation: The ...
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1answer
28 views

How can I find out the missing Markov transition probabilities given an incomplete transition graph?

I'm given a transition graph as shown below. I need to fill in the two missing probabilities. Is there a general method for doing this?
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17 views

Confounding Variable

Suppose researchers are doing some research between a group of 5 years old kids and a group of 10 years old kids. What would be the confounding variable here? Gender,height,income, aggression I ...
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1answer
21 views

Probability and statistics hw

On an outing of 100 students to a state park, 40 students brought neither a backpack nor a hat, 50 brought a hat, and 40 brought a backpack. If one of them was randomly chosen, find the probability ...
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1answer
83 views

Stochastic process: A bus with random numbers of passengers entering and exiting at each stop?

A bus with infinite capacity runs on an infinite route with stops indexed $n = 1, 2, 3,\dots$ The bus arrives empty at stop $1$. When the bus arrives at stop $n$, each passenger on it gets off with ...
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1answer
55 views

Sum of exponential random variables?

I am trying to find the PDF of $Y$, the sum of I.I.D. exponential random variables $X_1, ... X_n$ with $\lambda = 1$ and $n$ some known constant. So far, I have determined that moment-generating ...
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2answers
43 views

Operations on Random Variables and expected values

If we are given a random variable X with its own events and their respective probabilities, then how would you go about computing: $$ E(X^2) , E(2X+1)^2 $$ Moreover, how would you solve this problem? ...
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47 views

max distributions with IID with different parameters

Suppose that X,Y, and Z are exponentially distributed with mean 3,4, and 5 respectively. Further assume that they are independent. Find the expected value variance of the max(X,Y,Z). Answer; ...
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43 views

Logic behind Metropolis algorithm

I am using Metropolis algorithm to make a program for Ising model in Statistical Physics. In Ising model, we take a collection of spins with initial energy, say $E_i$, then we randomly flip one of the ...
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24 views

Why is $\hat y_i=\hat\beta_1-\hat\beta_2x_i$?

I'm trying to show that $\hat\sigma^2 =\frac{\sum\hat\epsilon^2}{n-2}$ is an estimator without biais and I started with: $$\hat\epsilon_i=y_i-\hat y_i$$ and my teacher suggested me to use the ...
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51 views

Sufficient statistic for $N(\theta,\theta^2)$ [closed]

Let $X_1,\ldots,X_n$ be a randon sample of the normal distribution with parameters $(\theta,\theta^2)$ How can I find a sufficient statistic for $\theta$? Is there an easy way to do it?
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33 views

Simulating Envelopes problem

I came across the envelopes problem (can be googled for more details) where you are given two sealed envelopes with one containing twice as much money as the other. You can pick one and then switch if ...
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3answers
57 views

Variance of the weighted sum of two random variables

I have a question with regards to the variance of the weighted sum of two random variables. Let's define $X$ as follows: $$X = \begin{cases} Y, &\text{ with probability } p\\ Z, & \text{ with ...
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1answer
125 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
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35 views

Order of A Random Sequence

Given that $\sup_{x\in\mathcal{X}}|\widehat{f}_n(x)-f(x)|=O_P(a_n)$ for $a_n\to0$, to characterize the order of $\sup_{x\in\mathcal{X}}\left|\frac{1}{\widehat{f}_n(x)}-\frac{1}{f(x)}\right|$, we can ...
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1answer
34 views

Hypothesis test: Testing for the median given a set of numbers

Suppose $X_1,...,X_n$ are froma continous distribution and we test whether the median of these observations is 100 against the alternative that the median is greater than 100. Let $Y$ be the number of ...
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18 views

Show that the first derivative at t=0 of $ln(M_{x}(t))$ is the expected value of X and its second derivative at t=0 is the Variance of X

Let $M_{X}(t)$ be a moment generating function of X. So far, I know that the first derivative of $\ln M_{x}(t)$ would be $M'_{X}(t)/M_{X}(t)$ and the second derivative would be $M_{X}(t) ...
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2answers
26 views

Probability and Expected Value as it approaches zero

I'm having issues understanding how to approach this question. Let $X_1, X_2, ... X_n$ be random variables in $(0,1)$ over some distribution. Prove that the following are equivalent. $\forall ...
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1answer
58 views

Statistics find E(x), the number of distinct elements in uniformly distributed pool of items

Question: Suppose there are Y types of balls in a bucket, which are normally distributed and independent. Hence the probability of picking one type out is $\frac{1}{Y}$. Let $x$ be the number of ...
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59 views

I don't need scientific notation! [closed]

I am looking for the probability of an event occurring 3 times with a decimal probability of .065 P(A) x P(B) x P(C) = 0.065 x 0.065 x 0.065 I am getting 2.74625E-4 I need this in decimal form not ...
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1answer
32 views

Is the p-value the probability that your null hypothesis is true?

I know that the p-value is the probability that the measurement takes a value that is at least as unlikely under the null hypothesis as the observed value. Furthermore, the null hypothesis is rejected ...
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1answer
32 views

“standard co-deviation”

This is a terminology/notation question. I swear I've seen covariance matrices written as \begin{bmatrix} \sigma_x^2 & \sigma_{x,y}^2 \\ \sigma_{y,x}^2 & \sigma_y^2 \end{bmatrix} Given ...
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Calculating Standard Deviation For Linear Combination Problem

So here's the problem: An airline charges the following baggage fees: 25 for the first bag and $35 for the second. Suppose 43% of passengers have no checked luggage, 38% have one piece of checked ...
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1answer
78 views

Find $E[(X − c)^+]$ when $X$ is normal with mean $\mu$ and variance $\sigma^2$

Find $E[(X − c)^+]$ when $X$ is normal with mean $\mu$ and variance $\sigma^2$. where $$ y^+ = \begin{cases} y & \text{if y>0} \\ 0 & \text{if y<0} \end{cases} $$ there is no ...
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How to find an upperbound on the time of depopulation of an urban enviroment

Given an urban area such as a shopping mall, is there a statistical model that estimates how often during a working day the area becomes depopulated and what is the maximum time of depopulation? By ...
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1answer
30 views

Why should I take this as the null hypothesis?

Let $X$ be the number of tails when throwing a coin ten times. $X$ has the binomial distribution with parameters $n$ and $p$. My nullhypothesis is $$H_0:p\leq 1/2$$ and the alternative $$A:p>1/2.$$ ...
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1answer
28 views

maximum and minimum distributions

Suppose that X is uniformly distributed on the interval [0,10] and suppose that Y=2X. Note that X is uniformly distributed on the interval of [0,20] Find the probability that min(X,20-Y)>7 What I ...
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2answers
54 views

$P(X_1 =\min\{X_1,X_2\})$ or $P(X_1<X_2)$

$X_i \sim EXP (k_i)$ I found the pdf of $Y=\min\{X_1,X_2\}$: $$f(y)=(k_1+k_2)e^{-y(k_1+k_2)}$$ I was told to use conditional probability for finding $P(X_1 = \min\{X_1,X_2\})$ Why can't I just use ...
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1answer
36 views

Joint PDF using dummy variables

Let X and Y are positive random variables whose joint PDF is given by $f_{X,Y}(x,y)=4e^{-2(x+y)}$. a) Find a PDF of X+Y b) Find a joint PDF of $U=\frac{X}{Y}$ and V=X c) Based on answer of (b), ...
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49 views

General equation for sampling without replacement probability

Looking at a preparatory exam, I'm a little dumbfounded by a question on probability. There are $19$ balls in a box: $5$ red, $3$ white, and $11$ blue. The question is: what is the probability of ...
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68 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim > N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
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96 views

definition of a sufficient statistic

The "normal" definition of a sufficient statistics is via independence of the pdf (conditional on the statistic) of the parameter $\theta$. The Fisher-Neyman theorem gives a nice characterization: ...
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24 views

Can you calculate confidence intervals for the population mean if the obtained sample is not normally distributed?

From my knowledge, if the obtained sample is approximately normally distributed, we can use t-tables to calculate the population mean confidence interval without knowing the population standard ...
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If I know that $P(N(B) = k \mid N(W)=n) = {n \choose k}p^k(1-p)^{n-k}$, and that $B \subset W$, how can I find $P(N(B) = k)$?

If I know that $P(N(B) = k \mid N(W)=n) = {n \choose k}p^k(1-p)^{n-k}$, where N(B) = number of events in set $B$, and that $B \subset W$, how can I find $P(N(B) = k)$ by itself? Is it feasible to ...
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In a joint distribution, is it true that $P(A_1, \dots A_n, A) = P(A_1, \dots A_n, A^{c})$ if $A = A_1 \cup A_2 \cup \dots \cup A_n$?

In general, if I have that $A = A_1 \cup A_2 \cup \dots \cup A_n$, and define $N(A_i)$ as the elements in $A_i$, and have $N(A_1)+\dots+N(A_n) = N(A)$, is this always true?: $$ P[N(A_1)=a_1 , \dots, ...
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131 views

Hipergeometric. Kids and candy.

Problem There are $15$ identical bags of candy each containing $20$ yellow, $15$ red, $5$ blue and $10$ green candies. $15$ children are each given their own candy bag and each randomly picks $12$ ...
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36 views

Given a cumulative distribution function of the form $P(X\leq x) = 1- e^{-\lambda x^3}$, is there a way to represent it using an exponential?

Given a cumulative distribution function of the form $P(X\leq x) = 1- e^{-\lambda x^3}$, is there any way to represent it in terms of an exponential or any other distribution? I've thought about ...
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1answer
20 views

Finding probability $P(x + y \le 1)$ given the joint pdf

So in the problem I was given the joint pdf $f(x,y) = x + y$, $0< x <1$, $0< y <1$, $0$ elsewhere. I am tasked to find $P( x+y \le 1 )$. My intuition was to $\int_0^1\int_{1-x}^1 (x+y) \, ...