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0
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1answer
29 views

writing down markov chain transition matrix

Question: An experimental animal can stay in room-A until 1 minute,and it can stay in room-B until 2 minutes. There exist deadly gases in room-C. One room among these three rooms is being randomly ...
1
vote
2answers
28 views

Test two groups with only sample statistics or without distributional assumptions

I have two sets of samples, A and B. I want to find whether the underlying mean (i.e. if the sample size was infinite) of A is greater than that of B, to a certain confidence (95%). There are two ...
9
votes
1answer
287 views

What is the difference between censoring and truncation?

Censoring: When an observation is incomplete due to some random cause. Truncation: When the incomplete nature of the observation is due to a systematic selection process inherent to the study ...
2
votes
2answers
40 views

Right censoring and Left censoring

Wikipedia gives the following definitions: Right censoring: a data point is above a certain value but it is unknown by how much. Left censoring: a data point is below a certain value but it is ...
0
votes
2answers
45 views

Lifetime or Failure Time

Lifetime / Survival time / Failure time : the time to the occurrence of event (always nonnegative) . Lifetime and Survival time can be synonymous . ...
4
votes
0answers
74 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
4
votes
1answer
61 views

Finding UMVUE of Bernoulli random variables

Given i.i.d. Bernoulli$\left(\theta\right)$ r.v.s $X_1, X_2, ...,X_n$, I'm asked to solve for the UMVUE of $\left(1−\theta\right)^2$ for the case $n=4$. I think that $\sum X_i$ is my sufficient ...
1
vote
2answers
131 views

Consequence of Multicollinearity

In case of perfect multicollinearity the predictor matrix is singular and therefore cannot be inverted . Under these circumstances, the ordinary least-squares estimator $\hat\beta=(\Bbb X'\Bbb ...
3
votes
0answers
47 views

Equal Prior Probablility and Linear Decision Boundary, a Simple Calculation Problem?

I get trouble in calculation on ML. I read one problem as follows: in a two class classification, with equal prior probability $P(C_1)=P(C_2)=0.5$ if the distribution of instance in classes be ...
4
votes
1answer
35 views

a challenge with linear classification and distance to origin? [on hold]

I ran into a problem, when studying on linear classification. my prof. says: in a linear classification $y=w_0+w_1x_1+w_2x_2$ that depicted on following figure, distance of origin to decision ...
1
vote
0answers
33 views

GARCH volatility forecast model in practice

I am self-studying GARCH models. I understand this is how it roughly looks: $$\sigma_t^2= \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^p \beta_i \sigma_{t-i}^2$$ and I understand ...
1
vote
3answers
57 views

Second moment from survival function

Let X be a non-negative continuous random variable with probability density function f(x). Let $$G(t) = \int_{t}^{\infty} f(x)dx$$ Show that$$E(X^{2}) = 2\int_{0}^{\infty} tG(t)dt$$ My thoughts: I ...
0
votes
0answers
11 views

Overlapping interval of a Poisson arrival process

Calls arrives according to a Poisson arrival process with rate lambda = 15. Find E(N(2,4]N(3,5]) My thoughts: E(N(2,4]) = E(N(3,5]) = lambda * t = 15 * 2 = 30 However, I cannot figure out the next ...
0
votes
0answers
44 views

Mean and Standard deviation [on hold]

The filling machine used by a dairy company to fill 1kg containers of yoghurt produces output which follows a normal distribution with mean 1030g (slightly more than 1kg) and standard deviation 20g. ...
-2
votes
0answers
25 views

PDF and return on stock [on hold]

i thought the answer is just 1 -P(x=<15) = 0.5 (1dp) http://i612.photobucket.com/albums/tt201/Ivanreyes/Screen%20Shot%202015-03-27%20at%206.33.13%20am.png
-1
votes
0answers
19 views

Equation in light of a given variance [on hold]

The total profit of a firm is given by Profit = Total Revenue – Total Cost Suppose that Total Revenue = 120Q and Total Cost = 20 + 50Q where Q, the quantity sold, is a (approximately) normal random ...
1
vote
1answer
20 views

Is pdf from power family distribution an exponential family?

In my statistics class we've been going over exponential families and sufficiency, which deviates from what's in the textbook. As such, now that I need to solve problems about exponential families I ...
0
votes
0answers
15 views

Does THE discordance test for outliers exist? [on hold]

I have looked but can't find 'the discordance test for outliers'. Does this specific test even exist? And if it doesn't, which test would you suggest to measure discordance for outliers?
3
votes
2answers
23 views

Definition of $X_t$ in the context of Stochastic process and Time Series

In the book An Introduction to Stochastic Modeling , Stochastic process is defined as : A stochastic process is a family of random variable(s) , $X_t$ , where $t$ is a parameter running over a ...
2
votes
0answers
19 views

How do I compute the CRLB to show if the estimator is indeed MVU

In system identification or parameter estimation, various input signals are used for exciting the process models. I am interested in parameter estimation of time series model using pseudo random ...
2
votes
1answer
33 views

Is there any difference between Random and Probabilistic?

It seems i can't directly say probabilistic and random are identical . But this is telling : random experiment is a probabilistic experiment. Is there any difference between Random and ...
2
votes
1answer
35 views

Implementing Gibbs sampler in R from posterior distribution

I am referencing a follow-up idea from something I posted earlier (Zero-inflated Poisson and Gibbs sampling, proofs and sampling). I want to implement the Gibbs sampler, by generating a large ...
-1
votes
0answers
20 views

Is Experiment and Trial synonymous?

Here they define : When we repeat a random experiment several times, we call each one of them a trial. But here they give the subtitle Experiment or Trial. ...
0
votes
2answers
46 views

Are Event and Outcome synonymous?

Outcome : An outcome is a result of a random experiment. Event : A single result of an experiment. Are Event and Outcome synonymous ?
0
votes
2answers
31 views

Discrete survivor function expressed in terms of hazard

Let $T$ can take on values $t_1,t_2,\ldots,$ with $0\le t_1\le t_2,\ldots,$ and let the probability function be $$f(t_j)=Pr(T=t_j),\quad j=1,2,\ldots$$ The survivor function is then $$S(t)=Pr(T\ge ...
0
votes
1answer
39 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 ...
0
votes
0answers
12 views

Joint Distribution and Marginal distribution [closed]

$$f(x,y) = \begin{cases} c(x+y) & x \gt 0 \text{ and } \ y \gt 0 \text{ and } (x+y) \lt 1 \\ 0 & \text{otherwise} \end{cases}$$ Show that $c=3$ I did this if $x\gt0, y\gt0, x+y\gt 1$ ...
2
votes
1answer
35 views

expected number of change points

A change point is defined as a term in a sequence of zeros and ones when a change happens. For example, in a sequence 1110110011 we have the following change points: $n_4,n_5,n_7,n_9$. Defining an ...
1
vote
1answer
32 views

How to find the long-run relationship using this regression (3rd time posted)

I know this is unorthodox but the exam is in 15 hours and if a question like this turns up I'll be unable to answer it. I've posted this twice already, the first time it was put on hold and the second ...
0
votes
0answers
20 views

Training and testing a Decision Tree Learner, can linear regression help obtain an optimal partition?

In class we are seeing decision tree learners and as a project I implemented from scratch my version of the one taught to us. I am using the famous Titanic Survival data (~1000 lines) for training and ...
1
vote
2answers
77 views

asymptotic distribution of a statistic

Say we have iid sample of size $n$ with $X_i \sim Exp(\lambda)$ and the task is to find asymptotic distribution of the statistic $$T_n := \frac{\bar{X}}{s}$$, where $s^2$ is the unbiased sample ...
0
votes
1answer
88 views

Finding a UMVUE for a specific function

Let $X_1....X_n$ be a random sample from the Poisson distribution with parameter $\lambda$. Find the uniformly minimum-variance unbiased estimator of a) $\lambda^3$ b) $E(e^X)=e^{\lambda (e-1)}$ ...
2
votes
0answers
18 views

Difficulties in interpreting the Equations in Parameter estimation for linear dynamical system

I have implemented the Kalman Smoothing with Expectation Maximization based on the Paper Parameter Estimation for Linear dynamical system. All notations are based on this paper. The model is an IIR ...
2
votes
1answer
84 views

Why $y_i$ becomes $(y_i-\overline y)$ in linear regression

Trying to figure out why $y_i$ becomes $(y_i-\overline y)$ in the below expression for finding $\widehat{\beta}$. Any help is highly appreciated.
4
votes
0answers
41 views

Does EM algorithm consistently estimating the parameters in Gaussian Mixture model?

I am studying the Gaussian Mixture model and come up with this question myself. Suppose the underlying data is generated from a mixture of $K$ Gaussian distribution and each of them has a mean vector ...
2
votes
1answer
58 views

How to solve the ball drawing problem?

Suppose a basket contains $B$ blue balls and $R$ red balls. Then, suppose I pick $N$ balls from the basket. What is the probability that I get At least $b$ blue ones and $r$ red ones At least $b$ ...
1
vote
1answer
22 views

Convergence and trend of Kalman Gain

I have implemented Kalman Filter for state estimation of AR(2) univariate model and wanted to plot the Kalman Gain. When implementing, I saw that Kalman gain for every sample is getting computed and ...
2
votes
2answers
124 views

Is a random variable Bernoulli? Is a proof available?

Suppose a die is tossed twelve times and each outcome is represented by a random variable $X_{i}$. Further define $Y_{i}$ for $i=2,...,12$ to take the value $1$ if $X_i=X_{i-1}$ and $0$ otherwise. ...
1
vote
1answer
40 views

Finding $Cov(2X+7, X^2 +3X - 12)$

So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess ...
3
votes
1answer
123 views

how to calculate a p-value for a null hypothesis given a z-score

I'm new to statistics and would love help with the following: Given an iid sample $X_1,...,X_{10}$ assume that $X_i\sim N(\mu,5)$. using the one-sample test we got a z-score of $2.2$. How do I ...
0
votes
0answers
21 views

OLS estimates consistent estimators of a pth order autoregression

I have a question from Hamilton's Time Series text. It is number 8.4: Consider a covariance stationary process of the form: \begin{equation} y_t = \mu + \sum_{j=0}^\infty \psi_j \epsilon_{t-j} ...
8
votes
1answer
213 views

If $X_1,X_2$ are independent beta then show $\sqrt{X_1X_2}$ is also beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
4
votes
2answers
132 views

How is the first column of the matrix orthogonal to all the others

$$ \mathbf{X}_{n\times(r+1)} = \begin{bmatrix} 1 & (x_{11}-\bar x_1) &\cdots & (x_{1r}-\bar x_r) \\ 1 &(x_{21}-\bar x_1) &\cdots & (x_{2r}-\bar x_r) \\ ...
0
votes
1answer
19 views

Learning statistics, cannot identify what a study is measuring

I have to describe the statistics of a study. I chose a study attempting to improve detection of mentally ill detainees by adding an additional questionnaire. The officers use the PISP and the study ...
0
votes
1answer
15 views

log multivariate normal differentiation (MLE)

I've come across a lot of explanations of how to differentiate the multivariate normal, but they all appear to skip the step that I'm stuck on. Here's what I've got so far. By logging and removing ...
0
votes
0answers
28 views

Sufficient Statistic for inverse Gaussian Distrobution

Let $X_1,...,X_n$ be a random sample from population with the pdf of the inverse Guassian distribution $f(x|\theta,\beta)=(\frac{\beta}{2 \pi x^3})^\frac{1}{2}e^{-\frac{\beta(x-\theta)^2}{2\theta^2 ...
0
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0answers
34 views

Matrix manipulation (weighted error term)

I have a question about matrix manipulation. I start off with a weighted error term. $$E_{D}(w) = \frac{1}{2}\sum_{n = 1}r_n (t_n - w^T \phi(x_n))^2$$ I differentiate, and set to zero to minimize ...
1
vote
0answers
50 views

Probability distribution of functions of random variables

A system will function as long as at least one of three components functions. When all three components are functioning, the distribution of the life of each is exponential with parameter ...
2
votes
1answer
42 views

how to get the critical region for a uniformly most powerful test for mean of normal?

I need help in understanding how to construct a uniformly most powerful test using the Neyman-Pearson lemma. Here is an excerpt in my text that I have trouble following: I have no idea how to get ...