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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Root Mean Square Error - How did he get this number?

So I am studying for a college final exam, and following a past exam paper at the moment. The lecturer has provided us with solutions to the previous years exam paper, not very clear in some cases ...
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1answer
20 views

Inverse of Score and Covariance/Variance matrix

I need to proof the following: $(A'A + B^{-1})^{-1}A' = BA'(ABA' + I)^{-1}$ Where B $\in{R}^{k \times k}$ is a variance matrix, $A\in R^{n \times k}$ of full rank. Unfortunately, I seem to get ...
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1answer
20 views

covariance of random variables

Suppose X, Y, W are independent random variables such that X ∼ GAM(2,3), Y ∼ N(1,4) and W ∼ BIN(10,1/4). Let U = 2X − 3Y and V = Y − W . Find cov(U, V ). I know that cov(U, V) = E(U, V) - E(U)E(V). ...
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39 views

Ratio of independant standard normal random variables.

I want to solve this question below. But I have no idea how to even start it. Any help would be appreciated.
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23 views

Hypothesis test to justify a claim

So I have a question regarding hypothesis tests where i have to justify a claim with statistical evidence. It is as follows: The average number of accidents in previous years in a city has been 15 ...
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1answer
26 views

A Property of Martingale of Sum of i.i.d. Random Variables

I am trying to solve the following problem: Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with finite mean. Let $F_n =\sigma(Y_1,...,Y_n)$. Let $\tau$ be a stopping time ...
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29 views

Why is $\mathbb{E}[(\bar{X})^{2}] \not = \mathbb{E}[X^{2}]$ in an iid distribution?

My work that is probably wrong somehow: $\mathbb{E}[(\bar{X})^{2}] = \mathbb{E}[(\frac{\sum_{i = 1}^{n}X_{i}}{n})^{2}]$ $ = \frac{1}{n^{2}} \mathbb{E}[(\sum_{i = 1}^{n}X_{i})^{2}]$ $ = ...
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17 views

Is there a notation to designate a random variable given it's distribution and vice versa?

Most of the times, I find it unnecessary and tedious to name the corresponding random variables of a given bunch of probability density functions. For example, one must write things like this even ...
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2answers
51 views

Expected score of a shot in archery

Here is the question: An archer fires at a circular target. Distances (in feet) from the target center are measured using Cartesian coordinates $(X,Y)$. Suppose each shot follows a random ...
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3answers
40 views

Uniform distribution and Exponential random variable mix

Here's the question Let X and Y be independent random variables, where X is uniformly distributed over (2,4) and Y is exponentially distributed with mean 3. Find the density of U=X/Y Here's what I ...
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0answers
31 views

Change of Variables: Ratio and Product of Two Uniform Distributed Random Variables.

Alright, here's a tough one that I can't figure out but I feel I've done quite a bit of work. Whether or not the work is right remains to be seen. Here is the question: Let $X_1,X_2$ be independent ...
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1answer
22 views

probability of running out of gas

So this is a two part question, but I figured out part B and I thought I had part A but it isn't working. Also, sorry if my formatting is not up to par, if anyone can reformat it if it looks bad, I'll ...
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2answers
36 views

Covariance problem involving and urn and some balls

Okay so here is the question - Suppose we draw five balls from an urn without replacement. The urn has $5$ red, $6$ black, and $5$ green balls. If $X$ gives the number of green balls drawn and and $Y$ ...
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1answer
36 views

Confusion about finding the CDF for this function .

Let $X$ be a continous random variable with the following PDF $$f(x) = \begin{cases} x& \mbox{for} \quad 0 \leq x < 1 \\ 2-x &\mbox{for} \quad 1 \leq x < 2\\ 0 ...
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1answer
18 views

Given the estimator find wheter

X has an uniform distribution on interval $(0,\theta]$ where $\theta$ is a positive parameter Given the estimator: $$T(X_1,X2, \ldots, X_n)=\frac{2}{n} \sum_{i=1}^n X_i$$ Find whether this estimator ...
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2answers
43 views

Determine who is the best seller

The numbers below show the number of lollipops Betty and Sharon each month for a total of 12 months or a year. Using the data and plot below, can you determine who is the bestseller? Would it be ...
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22 views

What is the domain of this random variable?

I've been self-studying Introduction to Statistical Learning. From page 16 of the book: "...suppose that we observe a quantitative response $Y$ and $p$ different predictors, $X_1$, $X_2$, ...
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1answer
15 views

How to estimate the max of a population using the normal distribution equation on a small sample

I recently watched a documentary on Mathematics. In the show they managed to estimate the weight of the largest fish that the fisherman was likely to of ever caught in his career just by analysing one ...
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16 views

Wald, LR and LM tests for Poisson distribution

Assume that $y$ is random variable which distributes: $$f(y)=\frac{e^{-\lambda}\lambda^{y}}{y!} $$ Where $\lambda=e^{\alpha+X\beta}$ and $\alpha$ and $\beta$ are unknown parameters and we assume that ...
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1answer
19 views

Beta distribution density function - integration problem

I want to calculate $EX$ of beta distribution given by certain formula (if necessary i will post it) i am stuck with an integral of this sort: $$\int_0^1\frac{\Gamma ...
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0answers
22 views

How to calculate survey bias due to preference for first answer?

I was recently given the results to a survey in which participants chose answers to questions they would be likely to randomly answer, and in which the survey population is known to have a preference ...
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1answer
40 views

Prove the consistency of Gamma distribution estimators

Given $X$ a random variable in a Gamma distribution, $f(x ; \alpha,\beta)$, and: $E(X) = \alpha \beta$ $Var(X) = \alpha \beta^2$ $\hat \alpha = $$\bar X \over \beta$ $\hat \beta = $$\frac {n \bar ...
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26 views

Why does Correlation Coefficient concern about the mean of the vector?

$$r = \frac {\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\sum_{i=1}^n(Xi-\bar X)^2} \sqrt{\sum_{i=1}^n(Y_i-\bar Y)^2}}$$ This is exactly the $\cos$ of degree of the angle between vector $X-\bar ...
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29 views

How was one derivied from the other?

In the geological paper entitled The power–law relationship between landslide occurrence and rainfall level by C. Li et al, a power-law cumulative probability distribution is derived. However, I don't ...
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33 views

Bias and variance of estimator

I have the following estimator, $E = 1/\bar{X}$ of $E = 1/\lambda$ where X is exponentially distributed with parameter $\lambda$. I'm trying to find the bias and variance of this estimator. For the ...
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1answer
12 views

Find the Sufficient Statistic for ß

The pdf is given by $$f(x|\beta) = \frac{1}{\beta+1} e^{-\frac{x}{\beta+1}}.$$ To find the sufficient statistic, we would find the Likelihood $L[\beta] = \frac{1}{(\beta+1)^n} ...
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1answer
43 views

Moment generating function help

I am not looking for just free handout answers but I am having extreme trouble with quite a few Prob and Stat problems. My instructor likes to do proofs in class and then assign application problems, ...
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1answer
35 views

Covariance in terms of variance [closed]

Is it true that? $$ Cov(X, Y) = \frac{Var(X) \; Var(Y)}{Var(X) + Var(Y)} $$ And if so, please prove or explain why this is so.
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1answer
16 views

Show that the estimator $\beta_1 = (X_1 + 3X_n + 4)/4$ is unbiased.

Suppose we are given an estimator $$\beta_1 = \frac{X_1 + 3X_n + 4}{4}$$ and the pdf given is $$f(x|\beta) = \frac{1}{\beta+1} e^{-\frac{x}{\beta+1}}.$$ and we have to show this is unbiased, i.e ...
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2answers
37 views

Choosing C so that c1Y1+c2Y2 is an Unbiased Estimator of Θ

Let Y1 and Y2 be two unbiased estimators of Θ let Var(Y1)= 4Var(Y2) and the correlation coefficient between Y1 and Y2 be -.5 find constants c1 and c2 such that c1Y1 + c2Y2 is an unbiased estimator of ...
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19 views

probability of max numbers and last numbers in sequence

If you have a sequence of 8 randomly generated real numbers in uniform distribution in the range of 1 - 10, where max = the two largest numbers in the sequence and last = the last two numbers in ...
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1answer
37 views

Expected number of rolls for fair die to get same number appear twice in a row?

We repeatedly roll a fair die until any number appear twice in a row. I want to find the expected number of rolls until we stop. I am thinking this is a geometric distribution, but how would I apply ...
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4 views

Risk of Maximum likelihood estimator is N?

Let $\mathbf{z| \mu} \sim \mathfrak{N}_N(\mu,I)$ where $I$ is N x N identity matrix. Total squared error loss to measure the error of estimating $\mathbf{\mu}$ by $\hat{\mathbf{\mu}}$, ...
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3answers
69 views

why is the geometric mean less than the logarithmic mean?

Can someone explain why the geometric mean is less than the logarithmic mean? $$\sqrt{ab} \leq \frac{b-a}{\log b-\log a} $$
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1answer
32 views

Find the Rejection Region

A drug claims to lower the blood pressure (by at least 10 mmHg) in 30% of all patients. A random sample taken shows a significant drop in blood pressure for only 20% of patients (n =35). Test the ...
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1answer
18 views

Force of mortality

The force of morality (hazard rate) at time $t$ is $\mu(t)$, so why does the expresion in the question denote the average force of mortality?
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9 views

Proof and precise formulation of Welch-Satterthwaite equation

In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows: Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ ...
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1answer
14 views

Wald Statistic and Confidence Region

I don't understand the conclusion in green here. I take it as to mean that if the hypothesis were correct the approximate 95% CR for $a \choose b$ is what has been underlined in red. However, ...
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1answer
25 views

Constant difference between two percentage numbers (nonlinear)

I'm trying to figure out the correct way of detecting the difference of values based on two percentages, because my terminology is poor here is an example: The difference between two values where ...
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0answers
5 views

Bottleneck Distance Significance?

Let $X$ be a smooth manifold and $f,g:X\rightarrow \mathbb{R}$ two real valued functions on $X$. Suppose we have two persistence diagrams $Dgm(f)$ and $Dgm(g)$ encoding the lifetime of $k$-dimensional ...
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22 views

What happens when we pick a Random sample?

Let $(\Omega, \mathcal{F} , \mathbb{P})$ be a probability space and $X:\Omega \to \mathbb{R}$ be a random variable. When we simulate or pick a random sample of size $n$ from $X$, are we picking ...
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1answer
16 views

Which variables to use in regression

If I have variables x1,x2,x3,and x4 that have correlation coefficients $-0.9, -0.5, 0.5,$ and $0.9$ to another variable y, what is the effect of choosing different combinations of them in a ...
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0answers
12 views

Multivariate extrapolation?

I believe this is basic statistics, but I don't know where to start looking. I have a function for a surface where $z = f(x,y)$. I also have a set of measurements $D$ at various $(x,y,z)$. There ...
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22 views

Intuition - Asymptotic Maximum Likelihood Estimator

Maximum Likelihood Estimation is quite clear to me when it is performed on finite sample sizes. The intuition of an obtained Maximum Likelihood estimate for given data $x_{1},...,x_{n}$, $n \in ...
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16 views

Expectation of chi-squared

Let $z \sim \boldsymbol{N}_N(0, (A+1)\textbf{I})$. The sum of squares $S=\left \| \boldsymbol{z} \right \|^2$ has a scaled chi-square distribution with N degrees of freedom, $S \sim (1+A)\chi_N^2$, ...
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1answer
35 views

Find MLE of $\frac{x+1}{\theta(\theta+1)}e^{\frac{-x}{\theta}}\Bbb I_{(0, \infty)}(x)$

Let $X_1,\ldots,X_n$ be a sample of size $n$, all with pdf $f(x)=\frac{x+1}{\theta(\theta+1)}e^{\frac{-x}{\theta}}\Bbb I_{(0, \infty)}(x)$ where $\theta > 0$. We will drop the indicator function ...
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1answer
25 views

What does it mean by Outcomes favour $H_0$?

Question How would you know what outcomes favour $H_0$ or $H_1$?
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6 views

Observed and expected Fisher information

I got some insecurity on having calculated the observed an expected Fisher information right in one of my study exercises. The distribution has a density of $\ f(x)=t*x^(t-1)$. For the Likelihood ...
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0answers
11 views

Marginal probabilities

I am stuck on the following problem of calculating marginal probabilities, which I have highlighted in yellow: Given the information below, how do we calculate$ p(X=0|w=\frac{1}{4}), ...
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32 views

Least Squares with Singular $AA^T$

Given the following system, find all least squares solutions: $\begin{bmatrix}1 & 2 & 3\\2 & 3 & 4\\3 & 4 & 5\end{bmatrix} \vec{x} = \begin{bmatrix}1\\1\\2\end{bmatrix}$ ...