Suppose that a discrete, random variable Y(with three possible outcomes) has the following distribution: prob(Y=1)=q, prob(Y=2)=p,...
Suppose that a discrete, random variable Y(with three possible outcomes) has the following distribution: prob(Y=1)=q, prob(Y=2)=p, and prob(Y=3)=1-p-q. A random variable of size 109 is drawn from this distribution and the random variables are denoted Y1, Y2,....Y109.
(A) Derive the likelihood function for the parameters p and q
(B) Derive the formulas for MLE of p and q
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Suppose that a discrete, random variable Y(with three
possible outcomes) has the following distribution: prob(Y=1)=q,
prob(Y=2)=p,...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; =...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...
[Q#2] (7pts) Suppose a discrete random variable Y has a Geometric probability distribution with probability of...
[Q#2] (7pts) Suppose a discrete random variable Y has a Geometric probability distribution with probability of success p (>0). Its p.d.f. p(y) is defined as P(Y = y) = p(y) = p (1-p)y-1 for y = 1,2,3, ... Verify that the sum of probabilities when the values of random variable Y are even integers only is 1-p. That is to find p(2) + p(4) +p(6) +.. 2 – p
Q1. Consider a random variable Y having probability density function otherwise. Given Yi, . . ....
Q1. Consider a random variable Y having probability density function otherwise. Given Yi, . . . , Yn, a sequence of г.г.d. observations on y 1. Determine the maximum likelihood estimator (MLE) of o. Denote this estimator, associated with a sample of size n, as d. Derive the score function, denoted by Sn (δ)-Olog ΓΤ:-1.fy (y|δ) Эд and show that it has an expected value of zero 3, Derive the information per observation. Эд and show that it is equal...
A discrete random variable has the distribution
A discrete random variable has the distribution, for n 1, 2, ...,. Random variables, {Xi:i=1,2,...}, do not depend on N and have the density fx (x) = 0.2e-0.2x for x > 0 and fx (x) = 0, elsewhere. Consider a random sum, 1. Find the expected value of Y. 2. Find the variance of Y. 3. Find the expected value of Y2
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What is the maximum likelihood estimator (b) Suppose p - 0.4 and n is unknown. What is the maximum likelihood estimator for p? for n? (Note: Since n is discrete you can't use calculus for this; just write the formula and use trial and...
PLEASE SHOW WORK Random variable ? has a discrete distribution below with all the possible outcomes:...
PLEASE SHOW WORK Random variable ? has a discrete distribution below with all the possible outcomes: X 1 2 3 5 ?(? = ?) 0.3 0.2 0.1 b a) Find the value b. b)Find the expected value of ?, ?(?). c) Find the variance of ?, ?ar(?). d) Find the standard deviation of ?, ?x.
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) -...
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) - fory - 1,2, ..., n. Show that p(y) satisfies the conditions of a pmf.
The discrete random variable has a uniform distribution. There are 12 possible values for the random...
The discrete random variable has a uniform distribution. There are 12 possible values for the random variable. One of the possible values is X = 5. P(X = 5) = _________________
The discrete random variable has a uniform distribution. There are 12 possible values for the random...
The discrete random variable has a uniform distribution. There are 12 possible values for the random variable. One of the possible values is X = 5. P(X = 5) =
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 =...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...
[Q#2] (7pts) Suppose a discrete random variable Y has a Geometric probability distribution with probability of success p (>0). Its p.d.f. p(y) is defined as P(Y = y) = p(y) = p (1-p)y-1 for y = 1,2,3, ... Verify that the sum of probabilities when the values of random variable Y are even integers only is 1-p. That is to find p(2) + p(4) +p(6) +.. 2 – p
Q1. Consider a random variable Y having probability density function otherwise. Given Yi, . . . , Yn, a sequence of г.г.d. observations on y 1. Determine the maximum likelihood estimator (MLE) of o. Denote this estimator, associated with a sample of size n, as d. Derive the score function, denoted by Sn (δ)-Olog ΓΤ:-1.fy (y|δ) Эд and show that it has an expected value of zero 3, Derive the information per observation. Эд and show that it is equal...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What is the maximum likelihood estimator (b) Suppose p - 0.4 and n is unknown. What is the maximum likelihood estimator for p? for n? (Note: Since n is discrete you can't use calculus for this; just write the formula and use trial and...
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) - fory - 1,2, ..., n. Show that p(y) satisfies the conditions of a pmf.
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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