Tagged Questions

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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1answer
14 views

Show that in this case $\rho \geq -\frac{1}{n-1}$

The bounds of correlation coefficient $\rho$ is shown to be $\pm 1$ in class. In many situations the bounds are sharper, i.e. they stay away from $+1$ or $-1$. Consider the random variables ...
-2
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0answers
13 views

Show that $Y_1, Y_2$ are independent N(0,1) random variables. [on hold]

Let $X_1, X_2$ be independent Uniform(0,1) random variables. Define $$Y_1=\,cos(2\pi X_1)\sqrt{-2\log(X_2)}$$ $$Y_2=\,sin(2\pi X_1)\sqrt{-2\log(X_2)}$$ Show that $Y_1, Y_2$ are independent N(0,1) ...
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0answers
15 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...
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0answers
12 views

Normal distribution-BMI [on hold]

The BMI index of Finish middle age men in finland follows the normal distribution with mean 27.5 and standard deviation 2.1. There are 642402 finish idle age men living in finland on November 9th ...
0
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0answers
18 views

Solving over-identified model in Mathematica

I am having trouble with using the "Solve" function for an over identified model in Mathematica. For a just identified model, where I have as many equations as I do unknowns the "Solve" function ...
0
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0answers
12 views

optimising likelihood function in R

I want to try and optimise the function below in R to find best estimates of $\tau^2$ and $\mu$. $L(\mu, \tau^2) = -\frac{1}{2}\sum\limits_{i=1}^k \mbox{log}(2\pi(\sigma_i^2 + \tau^2)) - ...
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0answers
13 views

How to check null hypothesis in Minitab without specific data?

I have been given a question that specifies a sample size of 50, sample mean of 3.05, standard deviation of .34, and desired mean of 3.2. The question asks whether or not the average mean is ...
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0answers
11 views

How to do the calculation of the error in positioning in 3D Space?

I'm mathematically calculating a position on given data in a program. But I need to validate that the calculated values against the original position value. If the Original position is Xi,Yi and Zi ...
5
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0answers
36 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $(\frac{X_i}{n^{\alpha}})_{i=1}^n$? More ...
1
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1answer
19 views

Rewriting Gaussian r.v. $Z$ as sum of two independent Gaussian r.v.

Suppose, $Z$ is Gaussian r.v. assume that it has mean 0 an variance 1. My question is can $Z$ be rewritten as \begin{align*} Z=\rho Z_1+(1-\rho)Z_2 \end{align*} where $Z_1$ and $Z_2$ are independent ...
4
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2answers
48 views

What is the expected value of min{|X|,|Y|}/max{|X|,|Y|} assuming X and Y are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and ...
1
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3answers
33 views

What is the probability that a fair coin comes up tail three out of four flips?

As the question asks, what is the probability that a fair coin comes up tail three out of four flips? I know the probability of getting tails on one flip is 1/2, but I'm not sure how to solve this for ...
0
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0answers
48 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
0
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1answer
19 views

Question about nonlinear regression

Speed: 25 35 45 55 65 Mileage: 20 24 26 24 20 The lease-squares line for predicting mileage from speed is mileage= 22.8 +(0*speed) Question: The correlation between mileage and speed is r=0. ...
4
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1answer
37 views

What are some good references on how probability theory got mathematically rigorous?

I am working on a term paper for an analysis course and I thought it would be interesting to talk about the connection between analysis and probability theory. Honestly, it would also benefit me a lot ...
0
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0answers
21 views

Determine how closely one set matches another. [on hold]

I am looking for ways to determine how closely one set matches another set. I'm devising a way to infer consumer preference for a business, but I have lots of ways I can tweak my model. My model will ...
1
vote
1answer
23 views

confidence inteverval $95\%$

how do I go about finding the $95\%$ confidence interval when I have $n=12, s_x=0.66$ and $u=35.72$ and also how many more samples would I need to reduce the "length" of the interval by half? so I ...
0
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0answers
16 views

Calculating the errors? [on hold]

Hollow cylinder of length l, inner and outer radii r1= 2.5+-0.3mm, r2= 5.5+-0.3mm, density= 7.88g/cm^(3), mass density= p, mass= (pi)xpxlx(r2^(2)-r1^(2)). What is the mass? Not sure how to approach ...
0
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0answers
20 views

Exponential distribution - Expected length of time

This was one of the test question that my teacher had put up. There is an electric board which contains 10000 bulbs, time out burnout for each bulb is an independent random variable following ...
0
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0answers
7 views

Distribution of sum of non central chi square random variable and a normal random variable

What is the Distribution of the sum of a non central chi square random variable and a normal random variable? Please do not remove question. Please post if you have a serious answer and explain if ...
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0answers
13 views

Multivariate Normal Variable (Homework)

I am trying to solve the following question: Let $(V, Z) ∼ MVN(0,I)$ (where I is the identity matrix) and let $Y=V+Z+1$. Find the distribution of $(1+Z,1-Y)$. I have found the distribution of ...
0
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1answer
15 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
0
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0answers
8 views

how can I Find a 95% credible interval for p using the Bayesian method with the uniform distribution as a prior for p?

When I have a RV X~Geom(p): $x\ Frequency\\ 1 7459\\2 1930\\ 3\ 463\\ 4\ 117\\ 5\ 22\\ 6\ 6\\ 7\ 2\\ 9\ 1$ This is what I am trying to do: Since p is a probability, I say that $ p\sim U[0,1]$ An ...
0
votes
4answers
54 views

Probability of 5 card hand

You have a $5$ card hand from randomly shuffled standard deck of $52$ cards: $X$ - Event that hand exactly contains $1$ spade $Y$ - Event that hand exactly contains $1$ ace How do I calculate $P(X ...
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0answers
12 views

implications of unimodality of distribution

Suppose $X$ is a continuous r.v. such that $X\ge0$ and its distribution is unimodal. What sort of consequences does that entail for $X$ in terms of its other properties (e.g., moments, etc.)?
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2answers
23 views

How to solve the PDF of lognormal distribution using the Normal Distribution

Let $X$ be $N (\mu,\sigma^2)$.Define the random variable $Y=e^x$ and find its probability distribution function. My solution is this, let $G(y)= P(Y\le y)=P(e^x\le y) =P(X\le ln y)$.Let ...
0
votes
1answer
21 views

Expected number of rolls to get 1, 2, 3 or 4

This is one of the interview questions. Suppose we roll the 6-sided die 7 times. All rolls are independent. What is the expected number of rolls to get either 1, 2, 3 or 4 ? How to solve it ? ...
0
votes
1answer
24 views

Distribution for random variable Z = Y1 - Y2

This was one of the interview questions. I did not know the answer. Question : Let Y1 and Y2 be two independent random variables where Y1 follows Normalpdf[x, -2, 5] distribution and Y2 follows ...
0
votes
1answer
32 views

values of p, so that$ f(0)\ge f(1)\ge…\ge f(10)$

Let $f(x)= {10 \choose x} p^x (1-p)^{10-x} $,$ x=0,1...,10$, zero elsewhere. Find the values of p, so that$ f(0)\ge f(1)\ge...\ge f(10)$. Here is my solution: $x=0 , {10 \choose 0} p^0 (1-p)^{10-0}; ...
1
vote
1answer
20 views

prove convergence almost surely

Let {$X_n$} be a sequence of i.i.d. random variables with $E[|X_1|]<\infty$, and $S_n=\sum_{j=1}^nX_j$. Show that if $E[X_1]\neq0$, then $$\frac{\max_{1\leq k\leq n} |X_k|}{|S_n|}\rightarrow 0 $$ ...
1
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0answers
16 views

Psyc - 295- stat method

Dave suspected that his hamster ate more than the average hamster (of the same breed) did. While the average hamster eats 900 g of food a day (SD=450), he monitored his hamsters for a month and found ...
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0answers
47 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
-1
votes
1answer
15 views

stats help with [on hold]

How large a sample should be taken if the population mean is to be estimated with $90\%$ confidence to within $83$? The population has a standard deviation of $893$. How do I do this?
0
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1answer
11 views

Finding density functions from conditional distribution

I'm currently taking a statistics course, but I'm having trouble with a specific concept, and hope this is a good place to ask. Essentially, for random variables $y_{1},y_{2}$, how do you get from ...
-2
votes
2answers
23 views

Correlation coefficient from regression line [on hold]

I encountered the below question in one of the interviews. Question : Suppose you have a random variable P and you define a new random variable $Q$ such that $Q = 2 - 3 P$. Calculate the value of ...
0
votes
1answer
24 views

Given 2 Random Variables, Please fill out the table

I am working on a problem and I have no clue where to start. I'm not sure what It is asking, or where to start. If you guys could give me the steps to take, show me what concepts are used, or a ...
6
votes
1answer
54 views

Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
0
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2answers
29 views

Probability - 5 card hand

Question is : You have a 5 card hand from randomly shuffled standard deck of 52 cards. P - Event that hand exactly contains one spade. Q - Event that hand exactly contains one ace. Calculate : a. ...
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0answers
10 views

Roots of Expectation of random variable raised to powers [on hold]

Given 1$\le$q$\lt$p And let X be a random variable such that $E(X)^p\lt\infty$
-1
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0answers
8 views

Does a joint PDF depend on whether one variable comes from a subset of the other?

If $\omega \in I_t$ ($\omega$ is a subset of $I_t$), then does $f(I_t,\omega) = f(I_t)$ where $f(\bullet)$ is a PDF? Clearly they are not independent. Is there a heuristic proof or counter proof out ...
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0answers
18 views

Deriving single linear regression parameters in terms of multiple linear regression parameters

Suppose the true population model is ln(wage) = B0 + B1(education) + B2(experience) + v (v is error term) Suppose the model is estimated as ln(wage) = B3 + B4(education) + u How do I calculate ...
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votes
1answer
19 views

Prove CDF is distributed Uniform$(0,1)$ [on hold]

Let $X$ be a random variable that has a Cumulative Distribution Function $F(X)$. Let $Y = F(X)$ be the random variable represented by the CDF itself. I seek to prove that $Y \sim Uniform(0,1)$.
0
votes
1answer
45 views

Students and Test Probability

Anna, Ben, and Chris write an exam that consists of only one question: "What is 26 times 26"?Both Anna and Ben give the correct answer with probability 9/10. Chris gives the correct answer with ...
0
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0answers
10 views

Expected Value of a random process

I can't for the life of me figure this out. Consider the random process x(t)=1/2 +(1/2)$\cos$(wt+$\theta$), where $\theta$ is uniform on the interval [0,2$\pi$] Calculate Expected value Calculate ...
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0answers
63 views
+100

How does 2D kriging interpolation work?

I have a grid of points Example ...
1
vote
1answer
27 views

When to expect normal distribution?

I was wondering when a normal distribution can be expected. I know that things like: heights of people size of things produced by machines errors in measurements blood pressure marks on a test ...
1
vote
2answers
18 views

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg.

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of ...
2
votes
0answers
22 views

Necessary sample size for difference between means

Not looking for a solution, just wonder what formula to use (and why?): Given population A and population B, where: standard deviation(A) = standard deviation(B) = $100$ If I select equal sample ...
0
votes
1answer
41 views

Method of moments, mle of $θ$, asymptotic variance [on hold]

This was on an exam we got back. I do not know how to do it but I would like to know now. Let $X_1,...,X_n$ be i.i.d random variables with the density function $$f(x\midθ)=(θ+1)x^θ, \qquad 0 ≤ x ≤ ...
0
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0answers
13 views

Math Statitics. Normal Distrubuton [on hold]

Studies shown that gasoline use or compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of ...