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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expected 1 norm of a normal vector

Given a normal vector $X$ such that $\mathbb{E}(X)=0$ and $Cov(X)=Id$, is it possible to get an expression for $$\mathbb{E}(\|A X\|_1)$$ where $A$ is a given matrix. I know that in dimension 1, we ...
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33 views

Calculate number of trials reaching $p_k$ probability for $k$ successes given the $p_t$ probability of each trial success

Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the ...
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1answer
26 views

Hypothesis testing: normal vs. non-normal

I have the following hypothesis testing problem: $$H_0:X=Y,\quad\text{vs.}\quad H_1:X=Y+Z$$ where $Y\sim\mathcal{N}(0,\sigma^2)$ and $Z$ is a random variable with non-normal continuous ...
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1answer
31 views

Understanding the value of standard deviation

I have two datasets $\{10,10,2,2\}$ and $\{13,7,0,4\}$. Now, when I compute standard deviation for both the sets, I get $4$ and $4.74$ respectively. My question: what is the significance of $4.74$ or ...
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23 views

Basic Asymptotic Theory book

I would love if someone can recommend me a book where Basic Asymptotic Theory is thoroughly covered and explained with some examples. I'm currently reading Econometric Analysis of Cross Section and ...
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1answer
26 views

joint pdf of $X$ and $Y$ find constant

$$f(x, y) = ce^{−x−y}$$ for $0 \le x \le y < \infty$, Calculate the value for $c$ that makes $f$ a valid pdf. how do you find $c$ when the domain of $x$ and $y$ both contain each other?
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14 views

An example where pearson is wildy different to spearman? [duplicate]

Im looking to spearman and pearson, and from what i understand spearman is better at looking at curves. Can i see an example of a small set of data (10 or less) where this difference is large.
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5 views

When to different hypothesis testing concepts

For hypothesis testing Am I right in saying that I use the Z test , if I know the two populations are independent and the two population’s standard deviations are known the pooled t-test, if the ...
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1answer
34 views

Variance of a random variable [closed]

How do you get the variance of a random variable $X$ where $X = \frac{1}{6}(A \cdot B)$ and where $A$ and $B$ are two independent random variables with variances $\sigma_A^2$ and $\sigma_B^2$, ...
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1answer
22 views

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $f_{X,Y}(x,y) = 6x$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $$f_{X,Y}(x,y) = 6x$$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$ I have the solution to this, but I don't understand it completely. Can some one ...
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2answers
22 views

Find $P(Y=5)$ for $Y=X_1+X_2+X_3$, where $X_i$ are mutually indpt Poisson R.V

In this problem, I am told that $X_1,X_2,X_3$ are mutually independent Poisson random variables with means $2,1,4$ respectively. I am also told to find the moment generating function for ...
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2answers
93 views

Suppose that $E[X^n] = 3n$. Find $E[e^X]$…

Suppose that $E[X^n] = 3n$. Find $E[e^X]$. Hint from my professor: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +···$ Not quite sure how to solve this problem, wouldn't $e^x$ go on exponentially. ...
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1answer
25 views

Use cdf to find expectation

I have a cdf for a $\mathbf {discrete}$ random variable, $X$, $$F_X(x)=1-(1-p)^{xn}$$ where $p\in(0,1)$, $n\in\mathbb N$, $x\in\mathbb N$ My thought is to use $$E[X]=\sum_{x=0}^\infty ...
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1answer
37 views

Let X be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$…

Let $X$ be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$ I would think $b$ is zero, making this an infinite sum but really not sure. Any help/direction with this problem is ...
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0answers
21 views

Show that if X is a continuous r.v. and it takes only positive values then $E(X) =\int_{0}^{∞}Z P[X ≥ t] dt $ [duplicate]

Show that if X is a continuous r.v. and it takes only positive values then: $$E(X) =\int_{0}^{∞} P[X ≥ t] dt$$ I am not really sure how to begin this proof. Any help or insight would be ...
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1answer
23 views

Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
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12 views

Find which categorical parameters have correlation with a given property [migrated]

I am a molecular biologist, so I apologize in advance, if the question is too basic. Example described below is for simplicity (I have a similar situation with my data to the one I describe below) I ...
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9 views

If a family of densities is not complete then is it necessary that there isn't any MVUE?

The question is about the truth of this statement: "If the family $\{f(x;\theta):\theta\in\Omega\}$ is not complete, then there doesn't exist any MVUE" MVUE is an abbreviation for "Minimum Variance ...
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9 views

Test the hypothesis that passing/failing this module is independent of the number of attendances

Marks ≥ 40% result in a pass, while marks < 40% result in a fail. Test the hypothesis that passing/failing this module is independent of the number of attendances The table for this question has ...
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13 views

Anti-intuition when finding statistical model of a random variable using Maximum Entropy Principal

I was trying to understand the Maximum Entropy Principal, and was calculating a very simple example, but ran into some confusion. Consider a random variable $X$, which can only take values $1,2$ and ...
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24 views

Consequences of fitting a regression model with an intercept term when it should be through the origin

Suppose a true model is $Y_i=\beta X_i +e_i$, where $e$ is the random error. Suppose instead we fit the model (using least squares) as $Y_i=\alpha_0+\alpha_1 X_i +v_i$, where $v$ is the random error. ...
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1answer
40 views

98% confidence interval

For this question from a large amount of data I have calculated that the mean is 44.22, the sample size is 100 and the standard deviation is 22.0773. From this I am asked to , make the 98% confidence ...
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23 views

Question on sufficient complete statistics proof and estimators of zero

I am trying to prove theorem 7.3.23 in Casella and Burger. Theorem: Let T be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$ be any estimator based only on T. Then ...
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7 views

Without homoscedasticity, is OLS still the best estimator (aka BestLinearUnbiasedEstimator…BLUE)?

Consider the Gauss Markov assumptions. Suppose we have a random sample $\lbrace x_n,y_n \rbrace_{n=1}^{N}$. Assume for a simple linear regression model $y_n = \beta_0 + \beta_1 x_n + \varepsilon_n$ we ...
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1answer
29 views

Compute the cumulative distribution function of the variable $R=\sqrt{X^2+Y^2}$

I've returned to the study of statistics after a long while and I'm trying to solve some problems. One of those is the next: Suppose $X$ and $Y$ are random independent variables with normal ...
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1answer
62 views

Forward price in Black Scholes Model

Recall that a forward contract on $S_T$ contracted at time $t$, with time of delivery $T$, and with forward price $f(t; T, S_T)$ can be seen as a contingent T-claim $X$ with payoff: $$ X = S_T - f(t; ...
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1answer
19 views

Hypothesis testabout the variability of 2 Samples

The hydrocarbon emissions are known to have decreased dramatically during the 1980s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), for ...
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1answer
26 views

What is the difference in how $\mathrm{R}^2$ and $\mathrm{R}$ values are interpreted?

In statistics, there is the $\mathrm{R}$ value for the product moment correlation coefficient and the $\mathrm{R}^2$ value for the coefficient of determination. In both cases they are described as a ...
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How is the $R^2$ value for exponential regression calculated if not by product moment correlation coefficient?

I am analysing some $x$ and $y$ values using Excel by plotting them on a graph and adding a line of best fit then using the equation for the lines of best fit. The exponential line of best fit has a ...
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11 views

Choosing between two exponential hypothesis

Problem. Let $P$ have density $e^{−x}$ on $[0,\infty)$ (so $P$ is a standard exponential distribution) and let $Q$ be the distribution of $X + 1$ where $X$ has distribution $P$. What is the maximum ...
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1answer
26 views

Hypogeometric Probabilities not adding to one

When I was computing the Hypogeometric formula probabilities I was having a little difficulty following the solutions manual. I attempted to compute the event where there was one Red Chip and One ...
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8 views

Why does the difference between $\mathcal{\hat{b}}$ and $\mathcal{b}$ converge to a normal distribution in a SLRM?

We are considering simple linear regression models in my econometrics class. We are given a random sample $\lbrace x_i,y_i \rbrace_{n=1}^\infty$ $$Y_i = b_0 + b_1 X_i + \varepsilon_i$$ where ...
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3answers
44 views

How does using one distribution as another's sample size affect variance?

How does using one distribution as another's sample size affect variance? For example, let's say I roll a 6-sided dice and record the number shown. Then, I roll 'that many' 6 sided dice more and ...
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28 views

X=the number of cards that are drawn until the 1st ace is chosen. Find E[X] [duplicate]

Consider a deck of $52$ cards. Let $X$ be the number of cards that are drawn until the first ace is chosen (e.g., if the first two cards are not an ace and the third card is an ace then $X=2$). Find ...
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1answer
12 views

One Sample Binomial or 2 Sample Binomial

So there is this one question, but I do not know what to use. (I believe it is 1 sample, can you plz tell me if I am right and wrong, and why?) The company Pepsi does a test to see what people like ...
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0answers
21 views

Suppose $f_{X,Y}(x,y) = \frac{3(4-2x-y)}{16}$ for $x>0$, $y>0$ and $2x+y<4$. Find $P(Y>2\mid X=1/2)$

Suppose $f_{X,Y}(x,y) = \frac{3(4-2x-y)}{16}$ for $x>0$, $y>0$ and $2x+y<4$. Find $P(Y>2\mid X=1/2).$ Not sure if I have the correct solution. Would someone let me know if this makes ...
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6 views

Estimator for covariance matrix

Suppose that I have a couple of units $X_{1}, \cdots, X_{n}$ to observe. What I have now are only $m$ observed samples for each units, which is $X_{i1}, \cdots, X_{im}$, $i=1, \cdots, n$ and the ...
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1answer
11 views

Generate an observation from a uniform (0,1) given a density function

Let X have density function $ \begin{cases} f(x) =(\alpha-1)/x^\alpha & \text{x>1} \\ 0 & \text{otherwise} \end{cases}$ , where $\alpha>1$ is a constant. How would one generate an ...
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1answer
28 views

derive the pdf for “difference of log-normal distributions”

Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ ...
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1answer
18 views

Variance of turning point - Time Series

I'm trying figure out why the variance for the turning point test is $V[T] = \frac{16n-29}{90}$. Given a sequence of points $\{y_i\}_{i=1}^n$, we define a turning point at time $i$ where $1 < i ...
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0answers
15 views

Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
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1answer
21 views

Type II error probability - is my solution correct?

When conducting a hypothesis test for a normal sample, with sample size = 5, with known standard deviation 0.02, where $H_0: μ = 1.12$, $H_1 : μ ≠ 1.12$ What is the probability of a type II error if ...
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2answers
75 views

Is a $90\%$ confidence interval really $90\%$ confident?

Let's say you are estimating a population proportion, which you model as binomial. One source of error already is using the normal approximation to the binomial when getting your critical values. But ...
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1answer
20 views

2 Discrete Random Variables Problems… [closed]

If I have 2 discrete variables, $X$ & $Y$ where X takes values in $\{ 1,2 \}$ and Y takes values in $\{0,1,2 \}$. The joint PMF is given by the table: I need to be able to determine the ...
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1answer
30 views

Differentiating a function with respect to two unknown.

Let, $$l(\mu, \sigma^2) = -(n/2)\log (2\pi)-n \log(\sigma^2)-\frac{1}{2\sigma^2}\sum_{i=1}^{n}(x_i-\mu)^2,$$ where $\mu$ and $\sigma^2$ are both unknown. How can I differentiate $l(\mu, ...
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1answer
40 views

Is there an interpretation of the hyper skewness?

Let $X$ be a random variable. The standardized $n$th moment of $X$ is defined as $$\frac{E[(X-\mathbb{E}[X])^n]}{\mbox{Var}[X]^{n/2}}. $$ Special cases are the skewness ($k=3$) and the kurtosis $k=4$. ...
0
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1answer
38 views

Calculating Probabilities with the PDF

I have that the PDF $$f(x) = \frac{33 - 4x}{75} \mathbb{1}_{\left\{2 \le x \le 7\right\}}$$ I need to be able to find $P(x > 4)$ & $P(x > 5.5|x > 4)$ but I don't know how. I thought I ...
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1answer
29 views

Independent and joint probability?

I got this question from my statistics teacher, but his answer made me confused. The question is this.. Given that A, B and C are three independent events such that P (A)=0.2 ,P(B)=0.6 ,P (C)=0.5, ...
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0answers
30 views

Statistical method used to compare internet speed

A cable provider received a complaint from a customer. The customer is complaining that the internet service is too low. The cable company servicer measured the internet speed at two locations in the ...
1
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1answer
33 views

Chi-square table

I'm new to using chi-square table, and I assume that it should not be hard to use but yet I have some problems with it. I am due to find the following in a chi-square table: $P(Y>5)$ where $Y$ ...