![1) [6 pts] Let Y be a Bernoulli random variable with success probability Pr (Y 1 )p, and let Y, Yn be iid draws from this distribution. Let p be the fraction of successes (1s) in this sample. (a) Show that p Y. (b) Show that p is an unbiased estimator of p. (c) (1-p)/n Show that var (p)-p](http://img.homeworklib.com/questions/ab67a420-7223-11ea-ab45-fb22ec655d04.png?x-oss-process=image/resize,w_560)
Homework Answers
Add Answer to:
1) [6 pts] Let Y be a Bernoulli random variable with success probability Pr (Y 1...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
1. Let X1, X2,... .Xn be a random sample of size n from a Bernoulli distribution...
1. Let X1, X2,... .Xn be a random sample of size n from a Bernoulli distribution for which p is the probability of success. We know the maximum likelihood estimator for p is p = 1 Σ_i Xi. ·Show that p is an unbiased estimator of p.
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known ΣΧ; is an unbiased estimator for p. that = n 2. Suggest an unbiased estimator for pa. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= ΣΧ,+2 n+4 is a biased estimator for p. 4. For what values of p, MSE) is smaller than MSE)?
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...
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 =...
Let X1, X2, .., Xn be a random sample from Binomial(1,p) (i.e. n Bernoulli trials). Thus,...
Let X1, X2, .., Xn be a random sample from Binomial(1,p) (i.e. n Bernoulli trials). Thus, п Y- ΣΧ i=1 is Binomial (n,p). a. Show that X = ± i is an unbiased estimator of p. Р(1-р) b. Show that Var(X) X(1-X (п —. c. Show that E P(1-р) d. Find the value of c so that cX(1-X) is an unbiased estimator of Var(X): п
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability fun...
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Question 3 [25] , Yn denote a random sample of size n from a Let Y,...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
1. Let X1, X2,... .Xn be a random sample of size n from a Bernoulli distribution for which p is the probability of success. We know the maximum likelihood estimator for p is p = 1 Σ_i Xi. ·Show that p is an unbiased estimator of p.
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known ΣΧ; is an unbiased estimator for p. that = n 2. Suggest an unbiased estimator for pa. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= ΣΧ,+2 n+4 is a biased estimator for p. 4. For what values of p, MSE) is smaller than MSE)?
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...
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 =...
Let X1, X2, .., Xn be a random sample from Binomial(1,p) (i.e. n Bernoulli trials). Thus, п Y- ΣΧ i=1 is Binomial (n,p). a. Show that X = ± i is an unbiased estimator of p. Р(1-р) b. Show that Var(X) X(1-X (п —. c. Show that E P(1-р) d. Find the value of c so that cX(1-X) is an unbiased estimator of Var(X): п
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Most questions answered within 3 hours.
-
how
does gravity affect the trajectory of projectile? what would be the
shape of the trajactory...
asked 6 minutes ago -
Two small plastic spheres are given positive electrical charges.
When they are a distance of 15.4...
asked 14 minutes ago -
An acidic solution containing gold ions is
electrolyzed, producing gaseous oxygen (from water) at the anode...
asked 29 minutes ago -
Assume that the population of Mexico is 128
million and that the population increases 1.01
percentannually....
asked 1 hour ago -
Can someone please help me add appropriate descriptive
comments to each line of code in the...
asked 1 hour ago -
Romeo wishes to throw a bouquet of flowers to Juliet, who is on
a second-story balcony,...
asked 2 hours ago -
Why is QE a controversial monetary policy tool.
A. It may lead to excessive inflation.B. By...
asked 2 hours ago -
Principles of Programming midterm study guide help!
1.)
______ Which of the following would reference the...
asked 2 hours ago -
A finite potential well has depth U0 = 2.78 eV . What is the
penetration distance...
asked 3 hours ago -
1. The bus bars of a power station are in two sections A and B
separated...
asked 3 hours ago -
Fiscal policy is the deliberate manipulation of taxes and
government spending to alter GDP, employment, inflation...
asked 4 hours ago -
evaluating an expression using only one digit and + and - as
operators ....3+5-1+7-5+8
-----------------------
stack...
asked 4 hours ago