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Exercise 11 (Delta Method) Suppose we want to estimate the variance of the Bernoulli distribtion T(p)...
Let X1,..., X, be a random sample from a Bernoulli(p) where p 1-(1-q*is the probability of...
Let X1,..., X, be a random sample from a Bernoulli(p) where p 1-(1-q*is the probability of a positive test when a test is conducted on a group of k subjects from a population with a prevalence rate of q. The estimated prevalence rate is l/k 9-1-(1-X) Using the Delta Method find the approximate variance of the estimator q.
Suppose X1,X2,…,Xn represent the outcomes of n independent Bernoulli trials, each with success probability p. Note...
Suppose X1,X2,…,Xn represent the outcomes of n independent
Bernoulli trials, each with success probability p. Note that we can
write the Bernoulli distribution as:
Suppose X1 2 X, represent the outcomes of n independent Bernou i als, each with success probabil ,p. Note that we can writ e the Bernoulǐ distribution as 0,1 otherwise Given the Bernoulli distributional family and the iid sample of X,'s, the likelihood function is: -1 a. Find an expression for p, the MLE of p...
Question 1. Suppose X1, X2, ..., Xn " Bernoulli(p). Define T = _, X; as the...
Question 1. Suppose X1, X2, ..., Xn " Bernoulli(p). Define T = _, X; as the sample sum and X = 1 2=1 X; as the sample mean. a) Find the exact distributions for T and X. b) Find the approximate distributions for T and X by CLT.
Let Xị, i = 1, ... be independent Bernoulli(p) random variables and let Yn = 1...
Let Xị, i = 1, ... be independent Bernoulli(p) random variables and let Yn = 1 1–1 Xį. (a) Use CLT to derive an approximate distribution for Yn. (b) Suppose that p + 1/2. Use Delta method to derive an approximate distribution for Yn (1 – Yn).
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0<p s an unknown parameter. Consider the parametric function T(p)-p + qe with q p. (i) Find a suitable unbiased estimator T for (p); (ii) Since the complete sufficient statistic is = Ση!Xi, use the Lehmann-Scheffé theorems and evaluate the conditional expec tation, E [I, I u-11]; (iii) Hence, derive the UMVUE for T(p) Hint: Try and use the mgf of the Xs appropriately.)
7.5.12 Suppose that X.., X, are iid...
3. In a Monte Carlo method to estimate T, we draw n points uniformly on the unit square [0, 1]2 and count how many poin...
3. In a Monte Carlo method to estimate T, we draw n points uniformly on the unit square [0, 1]2 and count how many points X fall inside the unit circle. We then multiply this number by 4 and divide by n to find an estimator of T (a) What is the probability distribution of X? b) What is the approximate distribution of 4X/n for large n? (c) For n- 1000, suppose we observed 756 points inside the unit circle....
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Suppose you want to estimate a particular population proportion p of “success”, say the proportion of...
Suppose you want to estimate a particular population proportion p of “success”, say the proportion of Cal Poly students who plan to go to Coachella this year. Consider two methods of collecting data. 1) Select a simple random sample of size n for a fixed, specified n. Let X be the count of successes in the sample. For example, select a sample of n = 30 students, and say for the selected sample X = 3 students plan to attend...
Suppose we conduct independent Bernoulli experiments with probability of success p once every hour. We track...
Suppose we conduct independent Bernoulli experiments with probability of success p once every hour. We track the number of successes over time. Let T = {1, 2, 3, . . .}. (a) Define the state of this process at time t, Y (t). (b) What is the state space at time t? (c) What distribution would each Y (t) have? (d) How are the random variables X(t) (from the Bernoulli process) and Y (t) related? (e) What would a plot...
IN EXCEL SO I CAN SEE THE FORMULAS USED. Open the Excel program Suppose we want...
IN EXCEL SO I CAN SEE THE FORMULAS USED.
Open the Excel program Suppose we want to know the standard normal distribution up to the given value. That is P(Z <a). Lets use Excel instead of the standard normal distribution table In cell A1 type mean. In cell B1 type 0. That means that we are letting mean be zero. click on cell A2 and type SD. In cell B2 type 1. That means that we are letting standard deviation...
Let X1,..., X, be a random sample from a Bernoulli(p) where p 1-(1-q*is the probability of a positive test when a test is conducted on a group of k subjects from a population with a prevalence rate of q. The estimated prevalence rate is l/k 9-1-(1-X) Using the Delta Method find the approximate variance of the estimator q.
Suppose X1,X2,…,Xn represent the outcomes of n independent
Bernoulli trials, each with success probability p. Note that we can
write the Bernoulli distribution as:
Suppose X1 2 X, represent the outcomes of n independent Bernou i als, each with success probabil ,p. Note that we can writ e the Bernoulǐ distribution as 0,1 otherwise Given the Bernoulli distributional family and the iid sample of X,'s, the likelihood function is: -1 a. Find an expression for p, the MLE of p...
Question 1. Suppose X1, X2, ..., Xn " Bernoulli(p). Define T = _, X; as the sample sum and X = 1 2=1 X; as the sample mean. a) Find the exact distributions for T and X. b) Find the approximate distributions for T and X by CLT.
Let Xị, i = 1, ... be independent Bernoulli(p) random variables and let Yn = 1 1–1 Xį. (a) Use CLT to derive an approximate distribution for Yn. (b) Suppose that p + 1/2. Use Delta method to derive an approximate distribution for Yn (1 – Yn).
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0<p s an unknown parameter. Consider the parametric function T(p)-p + qe with q p. (i) Find a suitable unbiased estimator T for (p); (ii) Since the complete sufficient statistic is = Ση!Xi, use the Lehmann-Scheffé theorems and evaluate the conditional expec tation, E [I, I u-11]; (iii) Hence, derive the UMVUE for T(p) Hint: Try and use the mgf of the Xs appropriately.)
7.5.12 Suppose that X.., X, are iid...
3. In a Monte Carlo method to estimate T, we draw n points uniformly on the unit square [0, 1]2 and count how many points X fall inside the unit circle. We then multiply this number by 4 and divide by n to find an estimator of T (a) What is the probability distribution of X? b) What is the approximate distribution of 4X/n for large n? (c) For n- 1000, suppose we observed 756 points inside the unit circle....
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
IN EXCEL SO I CAN SEE THE FORMULAS USED.
Open the Excel program Suppose we want to know the standard normal distribution up to the given value. That is P(Z <a). Lets use Excel instead of the standard normal distribution table In cell A1 type mean. In cell B1 type 0. That means that we are letting mean be zero. click on cell A2 and type SD. In cell B2 type 1. That means that we are letting standard deviation...
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