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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30 views

Determine the sample size needed to reject?

Determine the sample size needed to reject $H_0$ if the ratio is 3 times the hypothesized value of 2 variances, $\alpha = 0.05$, $\beta = 0.10$, $ R=3$. my answer is $$\text{samplesize}=\frac{R^2 ...
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0answers
27 views

Proof of Pearson's chi squared test

i was reading proof of this theorem on http://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec23.pdf They showed, that $\frac{v_j-np_j}{\sqrt{np_j}} ...
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0answers
28 views

Normal distribution of t

I have a population with 10000 individual, i took 10 to my samples. The values that i have inside $\bar{X}$ $$70.22502193, 70.26042017, 70.28977621, 70.2905717, 70.30113496, 70.32304453, ...
1
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2answers
27 views

MLE of MVN($\mu, \Sigma$)

I'm trying to find MLE of MVN($\mu, \Sigma$), i.e $N_k(\mu, \Sigma)$ with random sample $X_i, 1\le i \le n$. It was easy to get $\widehat{\mu}= \bar{X}$ and $\hat{\Sigma} = \frac{1}{n} \sum_i (X_i - ...
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0answers
26 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad ...
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0answers
35 views

Prove $P(X > s + t) = P(X > s)P(X > t)$

Say you have set $\{1,2,3,4,5,6,7,8\}$ and $s = 3, t = 4$ then $P(X>s) = 5/8$ and $P(X>t) = 4/8$, $P(X>s+t) = P(X>7) = 1/8$ $P(X>s)P(X>t)=20/64=5/16 \neq 1/8$ Where am I going ...
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0answers
10 views

relation between gaussian mixture models and maximum likelihood

I need some help understanding the relation between the maximum likelihood and Gaussian mixture models. I have seen that there is a relationship between the Expectation Maximization algorithms and ...
0
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1answer
40 views

$E[U^2]=0$ implies $Pr(U=0)=1$

DeGroot starts the proof to the Cauchy-Schwarz inequality with the assumption that $U$ and $V$ are random variables such that $E[UV]$ exists, then starts the proof with the following statement: If ...
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0answers
16 views

T student distribution

I have a population with 10000 individual, i took 10 to my samples. The values that i have inside $\bar{X}$ 70,22502193 70,26042017 70,28977621 70,2905717 70,30113496 70,32304453 70,32489699 ...
1
vote
1answer
102 views

Why doesn't Wilks 1938 proof work for misspecified models?

In a famous 1938 paper ("The large-sample distribution of the likelihood ratio for testing composite hypotheses", Annals of Mathematical Statistics, 9:60-62), Samuel Wilks derived the asymptotic ...
1
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2answers
16 views

X and Y are distributed by the same distribution

I know that if $X_1 \tilde~ Poisson(\lambda_1)$ and $X_2 \tilde~ Poisson(\lambda_2)$ then $X+Y \tilde~ Poisson(\lambda_1 + \lambda_2)$ Is this true with any distribution? Or is it merely because here ...
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0answers
54 views

$X<Y$ implies $E[X]<E[Y]$?

I just read a proof in a text that first established that $$(X-\mu_X)(Y-\mu_Y)\le \frac12\left[(X-\mu_X)^2+(Y-\mu_Y)^2\right]$$ then took the expectation of both sides of the inequality. ...
0
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3answers
29 views

DNA Sequence Distinct Way

we know The genetic code is based on the four nucleotides adenine (A), cytosine (C), guanine (G), and thymine (T). These are connected in long strings to form DNA molecule. with three A, one C, two G ...
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0answers
32 views

How would I analyze the accuracy of a model that predicts World Cup matches?

Say, someone made a bunch of predictions for each game between Team A and Team B, such that there's a predicted probability for each of the three possible outcomes adding up to $1.0$ : Team A winning, ...
0
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1answer
14 views

PDF, probability, variance, statis

Let X be normal with mean 1 and variance 4. LetY =2X+3. Find P(Y ≥ 0) I've solved E(Y) = 5 and Var(Y)=16. How do I apply this solve $P(Y\geq 0)$
1
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1answer
48 views

Rating system for 2 vs 2, 2 vs 1 and 1 vs 1 game

We play in table football in such configurations: 2 vs 2 players 2 vs 1 1 vs 1 Team (consisting of 1 or 2 players) wins the single game when they score 10th goal. I would like to introduce a ...
1
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1answer
33 views

(correction)Sampling distribution homework

I'm doing my homework and i dont have the answers, can someone say if its correct? "Let the population X, defined by X~Ber(p); Find The sampling distribution of an RV. X¯=1n∑ni=1Xi where Xi is the ...
4
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3answers
45 views

Question about English sentences in statistics?

Can somebody help me interpreting the red circled sentences in planer English? I understand "We view $y_i$ as a realization of a random variable $Y_i$ that can take the values of one and zero" but ...
3
votes
1answer
98 views

Linear combination of normally distributed variables

We know that if $X \sim N_p(\mu, \Sigma)$ then $a'X \sim N(a'\mu,a'\Sigma a)$ for and $a \in \mathbb{R}_p$. What I need to know is if the converse of this is also true. Can this be proved? Would ...
0
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2answers
31 views

Linear fit with Least Squares: what is going wrong?

I'm trying to do a Least Squares best fit for {1,2},{2,1},{3,3}. Unfortunately, my final formula continues to be wrong. As my final formula, I'm getting $y=\frac{1}{7}+\frac{13}{14}x$ - instead of - ...
2
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1answer
60 views

Expectation of Geometic distribution

I have the following question : It costs $30$ cents per day to keep pigeons. Let $N$ be the number of pigeons kept and suppose that $N$ has the geometric distribution $Pr(N=n) = \frac1{10} ...
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0answers
34 views

Solutions to exercises in Nelson's “An Introduction to Copulas”

I am paving my way through Nelson's "An Introduction to Copulas". The book has exercises (quite good actually), but no solutions. Does anybody have a solution manual for (some of those) exercises? ...
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2answers
57 views

Find out the value of $d$

If the mean deviation of number $1,\ 1+d,\ 1+2d,\ 1+3d,\ldots,1+100d$ from their mean deviation $255$ then $d$ equals to ? This was the question asked in AIEEE 2009. MY EFFORTS: ...
0
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0answers
31 views

normalize function -inf to inf to integrate to one.

I have a function of the probability distribution (method of k nearest neighbor). How I normalize this function , that the area under the graph is equal to unity. I mean it will make the prob from ...
2
votes
1answer
28 views

Markov chains by hand

If I have a starting point: $A_T=[0,1]$ at $T=1$ and a one step transition matrix of: $B=\left[ \begin{align} &\frac34 & \frac14& \\& \frac1{20}& \frac {19}{20} &\end{align} ...
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3answers
34 views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
0
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3answers
27 views

How can I do step-by-step calculations for the three equations to be equivalent to each other?

I am working on logistic regression, and have faced with the three equations below. I am told they are all equivalent. Why, how can I do the step by step calculations for them to be equivalent? ...
0
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2answers
27 views

Definition of Percentile (undefined notation)?

From Loss Models, 4th ed., by Klugman et al.: Definition 3.6 The $100p^{\text{th}}$ percentile of a random variable is any value $\pi_p$ such that $F\left(\pi_{p^{-}}\right) \leq p \leq ...
0
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3answers
50 views

$X_1$, $X_2$ independent implies $e^{tX_1}$, $e^{tX_2}$ independent

Suppose that $X_1$ and $X_2$ are continuous and independent random variables. How can I prove that $e^{tX_1}$ and $e^{tX_2}$ are independent?
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2answers
61 views

Existence of kth moment

Degroot makes a statement in his textbook: It can be shown that for every positive integer $k$, $$\int_{-\infty}^{\infty}|x|^ke^{-(x-3)^2}\,dx<\infty$$ Can someone show me how I might prove this ...
0
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1answer
56 views

A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

State any assumption(s) you make Well, I decided to draw a circle with a center at the origin of a Cartesian plane. It had radius r so it's coordinates on the axes were (0, r), etc. I then drew ...
0
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1answer
14 views

On a real line R points a,b are randomly selected such that -2<=a<=2 and 0<=b<=3. Find the probability that | a - b | > 1

Let's say that C is the set where |a-b|>1 So I suppose you could say plot it as coordinates where the x-axis (labelled a) is from [-2,2] and the y-axis (labelled b) is from [0,3]. Now |a-b| must be ...
0
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2answers
35 views

Testing if $X_1$ has an influence of $Y$

Consider you have the suspicion that $Y$ is influenced by two attributes $X_1$ and $X_2$: $$ Y=\theta_0+\theta_1X_1+\theta_2X_2+\theta_3 X_1X_2+U $$ The following data are given. Test ...
0
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1answer
13 views

Understanding details of PCA - variance

I am currently reading through Principal Component Analysis, Second Edition and came to small paragraph that I do not fully understand on page 5 (Section 1.1): "To derive the form of the PCs, ...
0
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1answer
12 views

Multiplying a binary predictor variable with another predictor variable.

Is it completely valid, to have an equation with a certain amount of variables, where two of the variables multiply each other? For example, I want to have an equation $Y = B_0 + B_1X_1 + B_2X_2 ...
3
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3answers
51 views

How to understand the variance formula?

How is the variance of Bernoulli distribution derived from the variance definition?
2
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1answer
23 views

Maximum likelihood estimator of a product of non-negative functions

Suppose that $a(\cdot)$ and $b(\cdot)$ are two non-negative functions such that $$f(x;\theta)=a(\theta)b(x)$$ is a probability density function for each $\theta > 0$. Find the maximum likelihood ...
3
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4answers
155 views

World Cup Group prediction probability

We have an office world cup bet where each person guesses the team that finishes 1st and 2nd from their qualifying group. E.g. A1: Brazil A2: Mexico B1: Netherlands etc... You get a point for ...
0
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1answer
22 views

Question about Logistic Regression - 5

Can somebody give me a clear explanation about logistic regression in bold below? Logistic regression can be binomial or multinomial. Binomial or binary logistic regression deals with situations in ...
2
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1answer
18 views

Adding stochastic variables random variables where one has time component

Based on GLS regression I have identified two random variables lets call them A & B ...
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1answer
24 views

Confidence Interval for Pivot

I don't follow the step highlighted in green.
0
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1answer
29 views

What are the purposes for all of the distributions?

I want a list of all of the main distributions and their application: Example entry: Poisson$(\lambda)$ - Models number of events in a given amount of time or space. Thank you, community answer ...
0
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2answers
30 views

Finding covariance between profit and quality

The quality $X$ of an item is uniformly distributed on the interval $[0,1]$ and the profit $Y$ is given by $Y = X^5$. Find the covariance between $X$ and $Y$ . Can someone interpret this question ...
2
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3answers
38 views

standard deviation of x= 121 divided by 121

$\{X_1, X2, \ldots, X_{121}\}$ are independent and identically distributed random variables such that $E(X_i)= 3$ and $\mathrm{Var}(X_i)= 25$. What is the standard deviation of their average? In ...
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2answers
23 views

Probability of second ball drawn not knowing the first with no replacement

I have 50 balls in a bag, 34 green and 16 blue. First person draws a ball but does not show it to me. Then I get to draw what are my chance of drawing a blue ball? This must be conditional probability ...
0
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0answers
50 views

World cup birthday paradox for two pairs

The BBC report on the world cup squads The birthday paradox at the World Cup shows the 50:50 prediction for there being 16 teams out of the 32 that meet the pair birthday criteria. However my ...
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0answers
12 views

Method of power law estimation rejects more accurate fits more readily as implausible than less accurate ones

The algorithm: The standard algorithm to estimate power laws and assess the goodness of fit (or rather the plausibility of the fit) is the one following Clauset, Shalizi, and Newman, ...
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1answer
32 views

Hypothesis testing: Test statistic, P-value and significance levels

A manufacturer claims his light bulbs have a mean life $μ = 1800$ hours. A consumer group tested a random sample of $n = 250$ bulbs and found them to have a sample mean life $\bar{x} = 1790$ hours ...
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1answer
13 views

Reference for Time Series and Linear Filters

I am reading "Time Series, Data Analysis and Theory" by David R. Brillinger. I am sorry but it seems to me that the book is quite sketchy and loose about notations. For example in the first chapter ...
0
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1answer
20 views

estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am ...