If we observe y0 as the value for a geometric random variable Y, P(Y = y0)...
If we observe y0
as the value for a geometric random variable
Y, P(Y = y0)
is maximized when p = 1/Y0.
The maximum likelihood estimator for p is
1/Y
(note that Y is the geometric random variable, not a
particular value of it). Derive E(1/y).
E (1/y) =
Homework Answers
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If we observe y0
as the value for a geometric random variable
Y, P(Y = y0)...
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1...
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1-p), x = 1.2. . . . , 0 < p < 1. Recall that E(x) (1-p) (a) Find the Fisher information I(p). (b) Show that the Cramer-Rao inequality is strict e) Let XX ~X. Find the maximum likelihood estimator of p. Note that the expression you find may look complicated and hard to evaluate. (d) Now modify your view by setting μ T1p such that...
N (,02). We 7. A positive random variable Y is said to be a lognormal random...
N (,02). We 7. A positive random variable Y is said to be a lognormal random variable, LOGN(1,02), if In Y assume that Y, LOGN(Mi, 0), i = 1,...,n are independent. [5] (a) Find the distribution of T = II Y. [4] (b) Find E(T) and Var(T) 5) (c) If we assume that Hi = ... = Hn and oi = ... = on what does the the successive geometric average, lim (IIYA), converge in probability to? Justify your answer....
1. Consider a variable y = θ+e where θ is an unknown parameter and e is a random variable with me...
1. Consider a variable y = θ+e 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 in y-Derive the least squares estimator for θ. For full credit you must check the 2nd order condition. (c) Can this estimator () be described as a method of moments estimator? (d) Now suppose e is independent normally distributed with mean 0 and...
A discrete random variable X follows the geometric distribution with parameter p, written X ∼ Geom(p),...
A discrete random variable X follows the geometric distribution
with parameter p, written X ∼ Geom(p), if its distribution function
is
A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with me...
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...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u,...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
Need help with this Problem 4 A discrete random variable X follows the geometric distribution with...
Need help with this
Problem 4 A discrete random variable X follows the geometric distribution with parameter p, written X ~Geom(p), if its distribution function is fx(x) = p(1-p)"-1, xe(1, 2, 3, . . .} The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that Ix(z) is indeed a probability inass function, i.e., the sum over all possible values of z is one...
(2) Let Y be a binomial random variable with parameters n and p. Remember that We...
(2) Let Y be a binomial random variable with parameters n and p. Remember that We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n借)(1-n) (a) Find the expected value of this estimator (b) Find an unbiased estimator that is a simple modification of the proposed estimator
We have seen that the geometric distribution Geo(p) is used to model a random variable, X...
We have seen that the geometric distribution Geo(p) is used to model a random variable, X that records the trial number at which the first success isachieved after consecutive failures in each of the preceding trials ("success" and failure being used in a very loose sense here). Here, p is the success probability in each trial. We described the geometric distribution using the probability mass function: f(X)(1- p)*-1p, which computes the probability of achieving success in the xth trial after...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y<...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1-p), x = 1.2. . . . , 0 < p < 1. Recall that E(x) (1-p) (a) Find the Fisher information I(p). (b) Show that the Cramer-Rao inequality is strict e) Let XX ~X. Find the maximum likelihood estimator of p. Note that the expression you find may look complicated and hard to evaluate. (d) Now modify your view by setting μ T1p such that...
N (,02). We 7. A positive random variable Y is said to be a lognormal random variable, LOGN(1,02), if In Y assume that Y, LOGN(Mi, 0), i = 1,...,n are independent. [5] (a) Find the distribution of T = II Y. [4] (b) Find E(T) and Var(T) 5) (c) If we assume that Hi = ... = Hn and oi = ... = on what does the the successive geometric average, lim (IIYA), converge in probability to? Justify your answer....
1. Consider a variable y = θ+e 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 in y-Derive the least squares estimator for θ. For full credit you must check the 2nd order condition. (c) Can this estimator () be described as a method of moments estimator? (d) Now suppose e is independent normally distributed with mean 0 and...
A discrete random variable X follows the geometric distribution
with parameter p, written X ∼ Geom(p), if its distribution function
is
A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability...
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...
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
Need help with this
Problem 4 A discrete random variable X follows the geometric distribution with parameter p, written X ~Geom(p), if its distribution function is fx(x) = p(1-p)"-1, xe(1, 2, 3, . . .} The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that Ix(z) is indeed a probability inass function, i.e., the sum over all possible values of z is one...
(2) Let Y be a binomial random variable with parameters n and p. Remember that We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n借)(1-n) (a) Find the expected value of this estimator (b) Find an unbiased estimator that is a simple modification of the proposed estimator
We have seen that the geometric distribution Geo(p) is used to model a random variable, X that records the trial number at which the first success isachieved after consecutive failures in each of the preceding trials ("success" and failure being used in a very loose sense here). Here, p is the success probability in each trial. We described the geometric distribution using the probability mass function: f(X)(1- p)*-1p, which computes the probability of achieving success in the xth trial after...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
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