Suppose that a random variable X has a Poisson distribution for which the mean θ is unknown (θ > 0). Find the Fisher information I(θ) in X.
Suppose that a random variable X has a Poisson distribution for which the mean θ is...
Suppose the random variable x has a Poisson Distribution with mean μ = 7.4. Find the standard deviation σ of x. Round your answer to two decimal places.
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Question 3 Suppose that the random variable X has the Poisson distribution, with P (X0) 0.4. (a) Calculate the probability P (X <3) (b) Calculate the probability P (X-0| X <3) (c) Prove that Y X+1 does not have the Polsson distribution, by calculating P (Y0) Question 4 The random variable X is uniformly distributed on the interval (0, 2) and Y is exponentially distrib- uted with parameter λ (expected value 1 /2). Find the value of λ such that...
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Suppose that X 1 has a Poisson distribution with mean 2, X 2has a Poisson distribution with mean 3 , X 3 has a Poisson distribution with mean 5 and that X 1 , X 2 and X 3 are independent. Define Y = X 1 + X 2 + X 3. Determine the moment-generating function for Y.
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with mean zero. (a) What is the expected value of y? (b) Suppose you draw a sample of yi yn. Derive the least squares estimator for θ. For full credit you must check the 2nd order condition c) Can this estimator (0) be described as a method of moments estimator? (d) Now suppose є is independent normally distributed with mean...
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is defined by de Show that, under some suitable regularity conditions, the variance of any unbi- ased estimator θ of θ is then bounded by the reciprocal of the Fisher information Var | θ 1(8) Note that the suitable regularity conditions, which are not specified here, allow the interchange of...
1) Suppose x has a Poisson probability distribution with mean 4.84. Find standard deviation. 2)Assume that x has a Poisson probability distribution. Find P(x = 6) when population mean is 1.0. 3)Assume that x has a Poisson probability distribution. Find P(x < 3) when population mean is 4.5