Estimate the number of goldfish

Reference no: EM131019735

1. Suppose that a family of four wants to estimate the number of goldfish in its pond (there are only goldfish in the pond). Call this number N. The family decides to use a capture/recapture strategy for collecting data. They plan to capture a random sample of Afish, mark them so they can be identified later, and then release them back into the pond. After allowing sufficient time for these fish to circulate with the other fish, they will then capture a new, independent random sample of ngoldfish and observe how many of those ngoldfish are marked. Let the random variable Xrepresent the number of goldfish captured in the second sample that are marked as having also been caught in the first sample.

(a) Derive an expression for the method-of-moments estimator of N, in terms of A, n, and X.

(b) Now suppose that the family captures 10 goldfish in the first sample and 12 goldfish in the second sample, of which 5 had been marked. Determine the methodof-moments estimate of N.

2. Reconsider the previous problem. Again suppose that the family captures 10 goldfish in the first sample and 12 goldfish in the second sample, of which 5 had been marked.

(a) Write out the likelihood function for N, and produce a sketch (you may use software for the sketch, excel will work).

(b) Determine the maximum likelihood estimate of N(again, you may use software to choose the maximum likelihood value from part (a)).

3. Let Y1, ..., Ynconstitute a random sample from a population with probability density function g(y|?) = (?+ 1)(y?) for 0 1, where the parameter ?is greater than -1.

(a) Determine the maximum likelihood estimator of ?.

(b) Determine the method-of-moments estimator of ?.

4. Let the random variable Yibe the number of typographical errors on a page of a 400-page book (for i = 1, ... , 400), and suppose that the Yi0sare independent and identically distributed according to a Poisson distribution with parameter ?. Let the random variable Xbe the number of pages of this book that contain at least one typographical error. Suppose that you are told the value of Xbut are not told anything about the values of Yi.

(a) Determine the maximum likelihood estimator of ?through the following steps:

i. Identify the probability distribution of X.

ii. Write out the probability function of Xin terms of the parameter ?.

iii. Write out the likelihood function of ?and also the log likelihood function of?.

iv. Take the derivative of the log likelihood function with respect to?.

v. Set that derivative equal to zero and solve for ?.

(b) Suppose that the data reveal that 25 of the 400 pages contain a typographical error. What is the maximum likelihood estimate of ??

Reference no: EM131019735

Questions Cloud

What is minimum amount of time takes for transmitted signals : In June 2004, the Cassini-Huygens spacecraft began orbiting Saturn and transmitting images to Earth. The closest distance between Saturn and Earth is 746 million miles. What is the minimum amount of time it takes for the transmitted signals to tr..

New cutoff from the previous part : Using the new cutoff from the previous part, what is the probability that they will decide to have the machines recalibrated if the truth is that µ= 11.9? How do the probabilities in parts (b) and (d) compare? Why is this?

Prepare a comparative income statement with horizontal : For 2014, Wiglaf Company reported its most significant decline in net income in years. At the end of the year, C.S. Lewis, the president, is presented with the following condensed comparative income statement: Prepare a comparative income statement w..

Constitute a random sample from a population : Let Y1, ..., Yn constitute a random sample from a population with probability density function g(y|?) = (?+ 1)(y?) for 0 1, where the parameter ?is greater than -1.

Estimate the number of goldfish : 1. Suppose that a family of four wants to estimate the number of goldfish in its pond (there are only goldfish in the pond). Call this number N. The family decides to use a capture/recapture strategy for collecting data.a

What does each box in an orbital diagram represent : What are "valence electrons"? What are "core electrons"? What does each box in an orbital diagram represent?

State the pauli exclusion principle in your own words : State the Pauli exclusion principle in your own words. The Pauli exclusion principle is, in an important sense, the key to understanding the periodic table. Explain.

Draw an arrow showing the excitation of an electron : What experimental evidence is there for the electron having a "spin"? Draw an energy-level diagram that shows the relative energetic positions of a 1s orbital and a 2s orbital. Put two electrons in the 1s orbital.

What is the company''s mission statement : If the company does not have a mission or vision statement, what might be the effect of that?

User Account

All Pages