Likelihood Definition and 139 Threads

Hello PF'ers, I am doing an unbinned likelihood analysis where I am analyzing the ratio of two spectra: \[ \frac{S_{1}(E)}{S_{2}(E)} = F(E) \] and each spectra, \[ S_{1}, S_{2} \] has its own data set. My first idea was to take the function, \[ F(E) \] and divide by the integral of...

I need help on this problem, anyone know how to do it? Suppose you have n independent observations from a uniform distribution over the interval [𝜃1, 𝜃2]. a. Find the maximum likelihood estimator for each of the endpoints θ1 and θ2. b. Based on your result in part (a), what would you expect...

Consider a density family $f(x,{\mu})=c_{{\mu}}x^{{\mu}-1}\exp(\frac{-(\ln(x))^2)^2}{2}$ , where $c_{{\mu}}=\frac{1}{{\sqrt{2{\pi}}}}\exp(-{\mu}^2/2)$ For a sample $(X_{1},...,X_{n})$ fnd the maximum likelihood estimator and show it is unbiased. You may find the substitution $y=\ln x$ helpful...

Homework Statement The independent random variables X_1, ..., X_n have the common probability density function f(x|\alpha, \beta)=\frac{\alpha}{\beta^{\alpha}}x^{\alpha-1} for 0\leq x\leq \beta. Find the maximum likelihood estimators of \alpha and \beta. Homework Equations log...

Homework Statement 1. An experiment consists of giving a sequences of patients a risky treatment until two have died, and then recording N, the number who survived. If p is the proportion killed by the treatment, then the distribution of N is: P(N=n)=(n+1)(1-p)n p2 1)Find a general formula...

Hi all I am using the Kalman Filter with an EM algorithm, (schumway and stoffer). From my understanding the log likelihood should monotonically increase. In some instances however I obtain a decrease in the log likelihood, What can I infer from this? Has anyone experience of these...

Hello, I have a question to Maximum Likelihood Estimation. The typical form of MLE looks like this: X = Hθ + W. W is gaussion with N(0, C). θml = (HTC-1H)-1HTC-1X I think θml can only be calculated after a lot of measurements are made, that is, there are plenty of samples of H and X...

Running a computer script (included below) I was testing to see how long it would take to match two numbers when selected at random from within a range. To my surprise the percentage of possibilities explored before finding a correct answer decreased as i raised the range. Is this correct? It...

Homework Statement 1. Suppose the data consist of a single number X, and the model is that X has the following probability density: f(x|θ) =(1+ xθ)/2 for -1≤ x ≤1; =0 otherwise. Supposing the possible values of θ are 0 ≤ θ ≤ 1; find the maximum likelihood estimate (MLE) of θ, and find...

I'm going to start by asking about an example from class and then hopefully use that to work on the problem I need to solve. Here is an example: Let's say we have a multinomial distribution $x \sim M(n;.5+.25\theta,0.25(1-\theta),0.25(1-\theta),0.5\theta)$. The likelihood function of $\theta$...

Hello: I've three circles as seen in the image. Source: http://www.picpaste.com/Quantify-PwdfQxLB.png Within the innermost circle is a point. I'd like to determine the likelihood, or probability, that this point belongs to the innermost circle, the middle-circle, and outermost...

Homework Statement Homework Equations So far I have only worked on question 1, as I was not able to solve it. The likelihood ratio test statistic is defined as follows: λ = 2 Log(L(theta-hat)/L(theta-hat_0)) Where L is the likelihood function, the product of all the pdfs/pmfs, and theta-hat is...

Hi everyone, Im running the KF to learn parameters of a model, the log likelihood of the p(Y_{k}|Y_{k-1}), however decreases. Can anyone advise, does this mean my implementation is wrong or can this just be the case. Advice appreciated Thanks

Homework Statement Hi all, I missed the day of class where we went over likelihood functions, and I'm quite confused! For example, let's say I have n Xis, where each Xi ~ Gamma(a,b), where a and b are unknown. I want to find the likelihood function of a and b, but I don't think I really...

Homework Statement John wants to measure the distance from his home to his office, so he drives to work several times and measures the distance on his car's odometer. Unfortunately, the odometer records distance only to the nearest mile. (Johns odometer changes abruptly from one digit to the...

Hi, I try to teach myself Hidden Markov Models. I am using this text "www.cs.sjsu.edu/~stamp/RUA/HMM.pdf" as Material. The Introduction with the example was reasonable but now I have trouble in unterstanding some of the derivation. I can follow the math and use the formulas to get...

Homework Statement I have a couple of notation interpretation questions: 1) What does f_X(x|θ) represent in this case? The realization function of of our random vector X for some value x and a parameter θ (so that if our random vector has n random variables, its realization vector will be a...

As far as I know, the definition of likelihood functions is the probability of a given random variable result given some parameter (please correct me if I'm wrong). What kind of parameters are usually handled by likelihood functions? Population parameters? Statistical model parameters? Both?

1.Suppose that X~B(1,∏). We sample n times and find n1 ones and n2=n-n1zeros a) What is ML estimator of ∏? b) What is the ML estimator of ∏ given 1/2≤∏≤1? c) What is the probability ∏ is greater than 1/2? d) Find the Bayesian estimator of ∏ under quadratic loss with this prior 2. The attempt at...

Homework Equations L(x,p) = \prod_{i=1}^npdf l= \sum_{i=1}^nlog(pdf) Then solve \frac{dl}{dp}=0 for p (parameter we are seeking to estimate) The Attempt at a Solution I know how to do this when we are given a pdf, but I'm confused how to do this when we have a sample.

I'm looking at using AICc to compare a set of nested models and wondered about the following. First, my the errors in my n data points around the fit are not i.i.d. Gaussian (the data have an artifact that the models will not able to fit easily and the artifact introduces dependencies and a...

Hi, Below is my attempt at a comparison between the two above-mentioned methods of estimation. Does anything in the table lack in validity and/or accuracy? Should any properties, advantages/disadvantages be eked out? Any suggestions/comments would be most appreciated! MLE...

There are likely to be infinite, or at least countless universes out there. We haven't seen them because any signs of their existence, such as wavelengths along the EMS, may very well take trillions of years go get here, and also may very well be too faint to detect from the astronomical (excuse...

Hi, Wondering if there is any priorities one method has versus the other one and are there any specific cases where to use one vs.other? regards

Homework Statement Let Y1,Y2,...,Yn denote independent and identically distributed random variables from a power family distribution with parameters α and θ. Then, if α, θ > 0, f(y|α, θ)={αy(α-1)/θα, 0≤y≤θ; 0, otherwise. If θ is known, show that ∏i=1n Yi is sufficient for α. Homework...

Homework Statement suppose that n cylindrical shafts are selected at random from the production of the machine and their diameters and lengths are measured. it is found that N11 have both measurements within the tolerance limits, N12 have satisfactory lengths but unsatisfactory diameters...

is it possible to estimate all parameters of an n-observation (X1,...Xn) with same mean, μ, but different variances (σ21,σ22,...,σ2n)? if we assume that σ2i are known for all i in {1,...n}, what is the mle of of μ?

Hello, I'm currently studying under an integrated M.S. program, majoring in Physics (Math as additional subject), and later wish to do Ph.D in theoretical Physics (broadly in particle physics). I know that universities look at factors like GRE score, reference letters, GPA, etc. as criteria...

This IS wrong. This only holds true if one of your parents is HETEROZYGOUS dominant. Hh x hh => 50% chance of offspring having Huntington's disease. However, if one of your parents is HOMOZYGOUS DOMINANT .. HH x hh => 100% chance of offspring having Huntington's diseases. Simple Punnett...

Homework Statement The question is about how to combine to different samples done with 2 different methods of the same phenomena. Method 1 gives normally distributed variables X_1,X_2,...X_{n_1}, with \mu and \sigma^2_1 Method 1 gives normally distributed variables Y_1,Y_2,...,Y_{n_2}...

Homework Statement An observation X has density function: f(x,/theta)=6x/(t^3)*(t-x) where t is a parameter: 0<x<t. Given the single observation X, determine the maximum likelihood estimator for t. Homework Equations Included below The Attempt at a SolutionFor a sample size of n...

Homework Statement An observation X has density function: f(x,/theta)=6x/(t^3)*(t-x) where t is a parameter: 0<x<t. Given the single observation X, determine the maximum likelihood estimator for t. Homework Equations Included below The Attempt at a Solution For a sample size of...

Given f(x; β) = [ 1/( β^2) ] * x * e^(-x/ β) for 0 < x < infinity EX = 2β and VarX = 2(β^2) Questions: Find the Maximum likelihood estimator of β (I call it β''), then find Bias and variance of this β'' 1/ First, I believe this is a gamma distribution with alpha = 2. Is that right? 2/...

Homework Statement I have a set of data from the DAMA experiment in which a detector attempted to measure collisions with 'WIMP's [Weakly Interacting Massive Particles] as a candidate for dark matter. The detector records the time in days of a collision event. After binning the data and...

Homework Statement According to genetic linkage theory, observed frequencies of four phenotypes resulting from crossing tomato plants are in the ratio 9/16 + a : 3/16 - a : 3/16 - a : 1/16 + a. In 1931, J.W. MacArthur reported the following frequencies: Observed...

Homework Statement Suppose that data (x1,y1),(x2,y2),.?.,(xn,yn) is modeled with xi being non random and Yi being observed values of random variables Y1,Y2,...Yn which are given by Yi = a + b(xi-xbar) + σεi Where a, b, σ are unknown parameters and εi are independent random variables each...

Hello, I am a Bio informatician and encountered Likelihood while executing the Molecular data. I have used one software that is using the Hidden Markov Model in addition to the EM Algorithm and Viterbi algorithm. After calculations are done already, in addition to the output, it is giving me...

Currently utilizing very simple logic in determining the directional trend of a signal line, and was hoping someone might be able to offer a suggestion as to a more effective method of filtering false signals. As it stands the logic being used for determining the direction of a trend is if...

Background: I'm a senior at a large state university studying electrical engineering, and I will graduate in May. Currently, my GPA stands at 3.08. I am on track for a 4.0 this semester, which includes a graduate-level course in digital communications systems. By the time I graduate, I should...

Hey guys how are you? I have the following question: Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf f(x|b)= (a*b^a)/x^(a+1) where x>=b (1) Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show...

Hey I have some problems understanding my statistics homework. I am given a data set giving the number of calls arriving to different switchboards in three hours as well as the total phone call duration in minutes for each switch board. Something like i y_i t_i -------------- 1...

Hi everyone, This is not a homework question but something I thought of while reading. In the method of maximum likelihood estimation, they're trying to maximize the likelihood function f(\vec{x}| \theta ) with respect to \theta. But shouldn't the likelihood function be defined as...

Monotone Likelihood Ratios--Most Powerful Test This isn't really a problem, its more of a theory question. I'm having trouble understanding the reading in my textbook (Introduction to Mathematical Statistics, by Hogg Craig, and McKean). I'm looking at the section of using concepts dealing...

Hi all, In this question, I found that \Pr(X_i|\theta)=\frac{\exp(\theta x_i)}{1+\exp(\theta)} and I carry on with the likelihood being \frac{\exp(\theta \sum x_i)}{(1+\exp(\theta))^n} and so s=\sum x_i = Tn I need some help with part (c)...

I was wondering -- what are the chances of someone getting away with an EMP attack? What measures is the US taking to prevent such an attack? The way I see it there are three scenarios that are more likely to happen: a) the attacker attempts to fly into the United States, the US sees an...

I actually have two questions, both of which are on the same topic Homework Statement Consider X = number of independent trials until an event A occurs. Show that X has the probability mass function f(x) = (1-p)^x p, x = 1,2,3,..., where p is the probability that A occurs in a single trial...

Homework Statement Hi everyone, I'm working through some max likelihood questions and am badly stuck on this one. Please could you take a look at what I'm doing and tell me if I'm going in the right direction? Q. Team A and B play two games of soccer, each game having two halves of equal...

Homework Statement A bag contains sequentially numbered lots (1,2...N). Lots are drawn at random (each lot has the same probability of being drawn). Two lots are drawn without replacement and are observed to be X_1 = 17 and X_2 = 30. What is the MLE of N, the number of lots in a bag...

Homework Statement Lifetimes of components are Gamma distributed. The parameters of the Gamma are shape = a scale = λ The pdf is: f(x) = (λ^a).x^(a-1).e^(-λx)/Γ(a) In this case, it is known that a = 3. Obtain the MLE of λ. Homework Equations The Attempt at a Solution Hi...

Homework Statement Let X_{1}, ... , X_{n} be a random sample from f\left(x; \theta\right) = \theta x^{\theta - 1} I_{(0, 1)}\left(X\right), where \theta > 0. a. Find the maximum-likelihood estimator of \theta/\left(1 + \theta\right). b. Is there a function of \theta for which there...