Hint: Use substitutions y = 1 - x-(μ-n)and y' =x-(μ-n)2121(e) Describe the criteria which can be used to choose between these two estimates.B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, withprobability mass function: P(X = x) = p(1 − p), with x = 0, 1, 2, 3,..., andp € (0, 1].(i) Using the MGF M(t) = 1-1-p)et derive E[X] and Var[X].(ii) Find the Maximum Likelihood Estimator (MLE) for p.(b) Suppose X₁,..., Xn is a random sample from a Beta(01, 1) population, and Y₁,..., Ymis an independent random sample from a Beta(02, 1) population. We want to findthe approximate Likelihood Ratio Test for Ho: 01 = 0₂ = 0o, versus H₁ : 0₁ 0₂.To this aim:(i) Under the alternative hypothesis H₁ 0₁ 02, show that the MLE for ₁ and0₂ are:0₁Page 5 of 6nΣ log(x;)Recall, that the PDF of Beta(a, b) is fy (y):m00=En log(yi)[(a+b)r(a) (b)I(a) = (a − 1)! for all positive integers a and I(1) = 1-(ii) Under the null hypothesis Ho: 0₁ = 0₂ = 0o, show that the MLE for 0 is:-¹ (1-y)-1 andn + mΣ log(x;) + Σ log(yi)Turn the page over

The discrete probability distribution is the distribution that has variables that take an integer…

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.
With an American penny, the likelihood of getting H when it is spun on edge is 0.3. If X is the random variable where X(H ) = 1, X(T ) = −1, find the expected value E(X), the variance, Var(X), and express X in its standard form.
The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 0
Let X1,..., X, a random sample with distribution f(x; 0) = (0 + 1)æº, 0 -1. (a) Find the method of moments estimator of 0. (b) Show that the method of moments estimator is consistent. (c) Find the maximum likelihood estimator of 0. (d) takes value 0 = 1.2. Use the parametric bootstrap method to obtain a 95% revised bootstrap percentile confidence interval using the maximum likelihood estimate. Make sure to include a plot of the bootstrapped values and an interpretation of your confidence interval. Given a sample of size n = 500, suppose that the maximum likelihood estimate of 0
Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |
A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = g(1-q). The experiment was repeated n times and x1,x2,..., Xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â = n n+S n where S = x₁. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁. (d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3. (i) What is the maximum likelihood estimate â? (ii) Find an approximate 95% confidence interval for q. (iii) Before seeing the data, our probability distribution for q has mean 0.4 and standard deviation 0.2. Find values of ao and Bo corresponding to this belief. What is then the…
4. Consider a random sample {X¡, i = 1, 2, ...n} of size n from a distribution with pdf f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise. (a) Show that the maximum likelihood estimator (mle) ô of 0 is a function of E-1 In(1 + X¡) (b) Let Y = In (1+X), find pdf of Y and mgf of Y. %3D (c) Using result (b) to show that Z = E, In(1 + X;) has a Gamma distribution. Specify the parameters of the gamma distribution (hint: find mgf of Z) (d) For test Ho: 0=1 vs Hi : 0 > 1, we reject null hypothesis if (X1, ..., Xn) € C, where C= {(X, .., Xn), E-, In(1 + X;) > k}. Give an expression to obtain k for given significance level a and what is the expression of power when 0 =1 (no need to integrate the function).

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(a) Consider X₁,..., X₂ to be a random sample from the geometric distribution, with probability mass function: P(X = x) = p(1 − p)ª, with x = 0, 1,2,3,..., and p € (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)et (ii) Find the Maximum Likelihood Estimator (MLE) for p. derive E[X] and Var[X].