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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Statistics-find the maximum likelihood estimator

Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution Laplace'a $\textrm{Lapl}(y,z)$ whose density function is: $f(x)=(z/2)*e^{-z|x-y|}$,where ...
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15 views

Relations between two versions of PCA

I have seen two versions of Standard/Classical Principal Component Ananlysis. And I have no idea how they are related: Version 1 :Wiki. This is about solving the eigen vectors and eigen values. The ...
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11 views

Laplace transform of squared fading

Say we have a formula for Laplace $$\prod \mathbb E_h[\exp(-sP_jh_{xj}l(xj))]$$ If we want to find the $\mathscr L_{h_{xj}}$ then $$\prod[\mathscr L_{xj}\exp(-sP_jl(xj))]$$ the channel $h_{xj}$ is ...
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19 views

Calculating the probability of something given the hazard rate function?

Suppose that the life distribution of a lightbulb of brand A has hazard rate function $λ_A(t) = t^{3}$ , t > 0. What is the probability that a brand A lightbulb burns out in less than 2 years?
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25 views

How would I calculate this probability?

The time (in hours) needed to restore power to a building after a storm is exponentially distributed with parameter 1/3. What is the probability that the building is without power for more than 3 ...
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1answer
13 views

How do I find Sxx in a Simple linear regression model?

In a Simple linear regression model, I have only Sxy and Syy data with me. How shall I derive Sxx, linking Sxy and Syy based on first principles? I know the formulas separately. I want to find Sxx, ...
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15 views

Likelihood ratio test for Bernoulli Distribution. [on hold]

I'm trying to solve a problem: Assume that random variables $ X_1,...,X_n $ are i.i.d. with a Bernoulli distribution with a parameter $ p $. Construct the likelihood ratio test for $ H_0 : p = p_0 $ ...
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1answer
16 views

Clarifying the importance of the quantile function in probability theory

I want to cement my understanding of the quantile function in probability theory and here is the way I understand it. (1) We start off with some probability space $(\mathbb R, B = \sigma(\mathbb ...
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1answer
6 views

Find critical value for uniformly distributed data set.

I'm trying to solve a problem... Got all the theory in hands. Not sure where to begin though... I have just started learning hypothesis testing and this is one of the problems I found on-line. Can't ...
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1answer
16 views

Statistics Notation of probability distribution

I am studying statistics but i'm a little unsure on the notation and I was wondering if anyone knew what this means when describing a probablity distribution? $$ X(\Omega, A, P) ->R $$ What do ...
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19 views

expected first passage time of a simple random walk [on hold]

For a symmetric, simple random walk on $S={0,1,...,k}$ let $T=\min\{n \in \mathbb N\ | \ X_{0}=x\}$ and $a_{x}=E(T|X_{0}=x)$ show that $a_{x}$ satisfies $a_{x}=0.5a_{x-1} + 0.5a_{x+1} + 1$ for $x ...
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14 views

Classical Regression Model: Combining linearity and strict exogeneity

I am studying the Classical Regression Model for random samples. Hence consider the random sample $(y_i,\mathbf{x_i})$ Where: ...
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0answers
17 views

Comparing 2 result sets with a confidence interval

I have 2 sets of data about student test results as following: School X had 36 students with the average of 75.21538 School Y had 41 students with the average of 74.0857 I have to check the ...
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12 views

Mathematical Statistics (Confidence intervals) [on hold]

I'm having trouble solving this problem from text.
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19 views

Statistics help please [on hold]

The heights of adult men are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches. The door has a height of 72 inches. A. How large would the door have to be so ...
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2answers
19 views

Maximum possible value for the probability of an event.

Consider two events $A$ and $B$ with the following given conditions: $$P(A)=0.4\quad\textrm{and}\quad P(B)=0.7$$ The maximum and minimum value of $P(A\cap B)$ are? Note: $P(X)$ denotes probability ...
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1answer
28 views

Statistics probability question [on hold]

Assume that 63% of people at a convention have at least a bachelor's degree. If a sample of 60 people was taken from the convention, what's the probability that the percentage of the sample with at ...
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1answer
22 views

finding lambda in a poisson distribution [on hold]

the number of times a frog jumps in a 10 minute interval follows a poisson distribution. The probability the frog jumps twice is three times the probability that the frog jumps 0 times. a) what is ...
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19 views

The Empirical Rule [on hold]

For the year 2003, the SAT test scores were normally distributed with a mean of 1050 and a standard deviation of 150. If 1495 SAT scores are in the top 0.14%, estimate how many students took the SAT ...
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1answer
26 views

What is the probability that the first failure due to a solar-panel problem will occur after the 20th launch?

I have the answer to this but I could not figure out how to get to it. I used the Geometric Probability formula and added all of the probabilities up to 20 using $(1-p)^{(x-1)}p$ but I get a wrong ...
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1answer
30 views

Integrating the log-normal function

Compute $$F(t)=\int_0^t \frac{1}{\sqrt{2\pi}\sigma t} \exp\left[-\frac{1}{2}\left(\frac{\log t-\mu}{\sigma}\right)^2\right]\,dt; t>0$$ My Attempt: $u=\frac{1}{t}\Rightarrow du=-\frac{1}{t^2}dt$ ...
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2answers
18 views

Find the expectations of the largest and smallest order statistics $X_{(n)}$ and $X_{(1)}$ respectively. Uniform distribution

Suppose that $X_1,\cdots, X_n$ are independent random variables from uniform distribution on interval $(\theta_1,\theta_2)$, $\theta_2>\theta_1>0$. It is know that $T(X)=(X_{(1)},X_{(n)})$ is ...
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18 views

Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?     ...
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11 views

How do I “fit” a 2D circular Gaussian with random samples?

I have N random samples taken from a circular Gaussian distribution with a known mean, namely (0,0), but unknown standard deviation. For each sample, I only have x^2+y^2, not x and y individually. ...
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1answer
18 views

Conditional Expectations Given Sum of I.I.D.

Given that $X_1,...Xn$ are all identical independent random variables. $\mathbb{E}(X_1|\sum_{k=1}^{n}X_k)$ = ? I am unsure how to proceed on this one. I know the default relation: $\mathbb{E}(X|Y)$ ...
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1answer
15 views

what is Likelihood ratio test for one observation $X\sim\mathrm{Pois}(μ+c)$?

We have one observation $X\sim\mathrm{Pois}(\mu+c)$ where $c$ is a known constant. We wish to test $H_0: \mu=0$ vs $H_a: \mu>0$. Derive the likelihood ratio test for this problem. Do the test at ...
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22 views

Calculating covariance (PCA example)

What is $z_1$? We have $n$ observations of $p$ different quantities (variables). It's defined on page 6. Is it just a projection onto the vector $a_1$? $a_1$ is our new basis vector and thus $z_1$ is ...
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17 views

finding a distribution that fail regularity condtion?

I know that uniform distribution that fail regularity condition. Can you find another distribution that fail regularity distribution? thank you for helping.
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1answer
20 views

Statistics acceptance sampling plan [on hold]

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test $12$ tablets, then accept the whole batch if there is only one or none ...
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1answer
19 views

Statistics normal distrution probability

A vending machine dispenses coffee into 8 ounce cups. The amount dispensed into these cups is normally distributed with a.mean of 7.6 oz and a standard deviation of 0.4 oz. a) Estimate the ...
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1answer
18 views

Find the UMVUE of the range and mean parameter of a uniform distribution

Suppose that $X_1,\cdots, X_n$ are independent random variables from uniform distribution on interval $(\theta_1,\theta_2)$, $\theta_2>\theta_1>0$. It is know that $T(X)=(X_{(1)},X_{(n)})$ is ...
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8 views

Does the estimator of $P(X\leq 1)$ for Poisson distribution attain the Cramer-Rao lower bounf

Let $X_1, \cdots, X_n$ be a random sample from a Poisson distribution with mean $\mu$, $\mu>0$. (a) Find the UMVUE of probability $P(X \leq 1)$. (b) Does the estimator obtained in part (a) attain ...
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22 views

Inferences for Single samples: Hypothesis Testing

In order to test the accuracy of speedometers purchased from a subcontractor, the purchasing department of an automaker orders a test of a sample of speedometers at a controlled speed of 55 mph. At ...
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15 views

Find the p-value

I'm strugling tyring to find the p-value if i'm given a Ho Ha and z-statistic.. Also I need to consider if it is one side or two sides... So for example find the p-value: ...
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19 views

Sampling from conditional distiribution

I want to sample from condtional distribution given by formula: $$ P[T_{n+1} =t| T_n=t_n] = ...
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2answers
26 views

odds of having last card in 52 card deck be the Ace of spades [on hold]

My social club has a 50/50 drawing each week. We purchase tickets and if your ticket gets picked out of the hat, then you draw a card from a standard deck of 52 cards. If you get the Ace of Spades, ...
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2answers
48 views

Need help interpreting weird result for seemingly simple problem.

I am solving a problem and just can't wrap my had around the result I'm getting... Here it is: So my next step would be setting derivative equal to zero and solving for Theta... Are my calculations ...
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15 views

What is $\operatorname{Var}(x^y) x$ is normally distributed random variable?

I am trying to understand the $\operatorname{Var}(x^y)$. I thought that it is $E(x^2)^y-[E^2(x)]^n$ but I understand that I was wrong, or wasn't I?
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1answer
29 views

Positivity of pdf of sum of non-iid random variables

Suppose I have two random variables $X_i, i=1,2$ distributed on open subsets $U_i$ of a unit ball around $0$ in $\mathbb{R}^d$. Suppose $0\in U_i$ for every $i$. I assume that distribution of each ...
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1answer
24 views

Bayes vs frequentism and the fair coin

Suppose I have a coin, which I want to test for bias. My problem is: surely there's a philosophical problem with defining "bias". Let me illustate with an example. Firstly, I use a Bayesian approach, ...
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6 views

Discriminant Functions of two classes sharing same covariance matrix

How can i find the discriminant functions of two classes having same diagonal covariance matrix with different means? (their feature vector is two dimensional) Thank you!
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1answer
23 views

Given a uniform $X$ distribution how can I find distribution of $Y=\ln(x)$? [on hold]

If $X$ is distributed uniformly on $(0,1)$ what is the distribution of $Y$ if $Y=\ln(X)$?
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10 views

Log likelihood function for binary classification

I need help with this following task. There is a binary classification problem where each observation xn is belong to one of two classes (t = 0 and t = 1). The training data points are sometimes ...
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1answer
12 views

bias-variance trade off

Perhaps the most primitive (and yet very effective) way of estimating a density is by using a histogram. In a histogram, we can manage the 'precision' of the distribution by adjusting the number of ...
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estimation of the parameters of generative process modelling second-price-auction

The generative process: There are 2 entities (A,B) entity A - is the exchange performing second-price-auction entity B - is somebody who is trying to understand the distribution-of-the-value people ...
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1answer
16 views

To calculate variance, given conditional distribution

Let Y be an exponential random variable with mean $\frac{1}{\theta}$, where $\theta>0$. The conditional distribution of X given Y has Poisson distribution with mean Y. Then, the variance of X is ...
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26 views

Minimizing the length of the confidence interval for the unknown normal mean with known variance

Reading through my Stats textbook, the author briefly talks about the minimal length of the confidence interval for the unknown normal mean with known variance. He states that the confidence interval ...
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1answer
21 views

UMVUE for pdf $f_{\theta}(x) = \theta e^{-\theta x}, x>0$

Let $X_1,\ldots,X_n$ be a random sample from a pdf $f_{\theta}(x) = \begin{cases} \theta e^{-\theta x}, & x>0 \\ 0, & \text{otherwise} \end{cases}$, where $\theta>0$ is an unknown ...
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1answer
9 views

Calculating MGF for a random variable with pmf $P(X=x)=k\cdot( ^nC_{x})$

The pmf of a random variable X is given by $P(X=x)=k\cdot( ^nC_{x})$, $x=0,1,2,...,n$, where k is a constant. The moment generating function $M_X(t)$ is (A)$\dfrac{(1+e^t)^n}{2^n}$ ...
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11 views

Expectation of Multivariate Normal - help please

I am struggling to show why an equation is true. Help would be very much appreciated Given that x $\sim$ N($\textbf{m}$, $\Sigma$) [multivariate normal], how could you show that: E[($\textbf{x}$ - ...