Ask Question, Ask an Expert


Ask Statistics and Probability Expert

1) The t and F Distributions

The T random variable with v degrees of freedom has the distribution

1886_T and F distribution.jpg

Show that the random variable F = T2 has the F distribution with 1 and v degrees of freedom.

2) An Unbiased Estimate of σ

The chi distribution models the square root of a chi-square random variable:

                        Y= √x      X~ x2 v

(a) Find the pdf of the chi distribution.

(b) We know that the sample variance from a normal distribution follows a chi-square distribution

265_chi square distribution.jpg

Find the expected value of the sample standard deviation, s, and suggest an adjustment that would make it unbiased. You may find the following formulas useful:

2057_Expected value.jpg

3) A Catch-and-Release Estimate

A park has N raccoons of which 10 were previously captured and tagged. Suppose that 20 raccoons are captured. Find the probability that n = 5 of these are found to be tagged. Denote this probability by p(N).

(a) Find the value of N that maximizes p(N); this is called a maximum likelihood estimate. Hint: compare the ratio p(N)/p(N -1) to unity.

(b) Plot the maximum likelihood estimate of N against varying values of n, from 1 to 10.

4) Finite Population Correction Factor

For a finite population of size N with mean µ and variance σ2, it can be shown that the covariance of any two observations in a sample is


and that the sample variance is slightly biased

1226_biased sample variance.jpg

This means that the variance of the sample mean, when drawn from a finite population, includes a covariance term. This gives rise to the finite population correction factor. Use these relationships to show that the variance of the sample mean is

2482_mean variance.jpg

and that this is estimated by

813_Estimation of variance.jpg

5) Zero-Intercept Regression

Some phenomena follow a linear model which has an inherent zero response for a zero predictor, so the candidate regression model is


(a) Find the ordinary least squares estimator for β  by minimizing the sum of squared errors

535_least square estimator.jpg

(b) It seems reasonable that the variance around the regression line would increase as the predictor increases (think of the line swinging around the fixed "pivot" at the origin). If the error was normally distributed, this could give rise to the conditional distribution

1940_conditional distribution.jpg

Assuming this distribution, find the maximum likelihood estimator for β .

describe why this is called a ratio estimator.

(c) for the following data, estimate β using both your OLS and MLE estimators.

x  0.5 1.5 3.2 4.2 5.1 6.5

y  1.3 3.4 6.7 8.0 10.0 13.2

6) A Truncated Distribution

Students in several statistics classes were asked to complete a problemnaire. One of the quantities asked was the number of siblings a student had. This is a summary of the responses:

siblings    frequency

0                  4

1                 22

2                 22

3                 11

4                  8

5                  3

6                  3

12               1

20               1

6.1) Problem. Use the sibling data to estimate a distribution for the number of children in a family. Obviously, the number of children is one more than the number of siblings. However, there is a selection bias in this measurement; families with no children cannot be reported this way! Therefore the data follows a zero-truncated Poisson distribution (ZTPD)

1160_zero-truncated poisson distribution.jpg

(a) Find the MLE for the parameter λ of the ZTPD.

(b) What are the mean and variance of the ZTPD? Rather than directly calculating the moments, you might find it simpler to use the probability generating function

                                                  Gk(z) = E[zk]

and then take advantage of the properties of the pgf:

E[K] = G'k(z = 1)   V ar[K] = G"k(z = 1) + G'k(z = 1)- [G'k(z=1)]2

(c) Estimate the mean for the number of children and determine whether the data fits a ZTPD with that mean.

7) Estimating with Confidence The times to failure (in hours) for a sample of n = 30 backup generators are

7494.7       8801.7     9990.7      11277.7     10173.3      7746.8

9003.6       8242.9     4532.2      12541.8      6766.9       9898.9

8922.0       13429.8   17623.5    9135.6        6029.8       9038.7

20972.0     7605.1     5396.6      7528.2        10330.6     6475.4

12390.9     9857.0     7067.6      9704.2         5055.8      9942.4

(a) Find the mean time to failure and a 95% confidence interval (use a t distribution).

(b) Find a 95% confidence interval for the standard deviation (start with a chi-square interval for the variance).

Statistics and Probability, Statistics

  • Category:- Statistics and Probability
  • Reference No.:- M9290

Have any Question? 

Related Questions in Statistics and Probability

An employer interviews 9 people for 6 computer programming

An employer interviews 9 people for 6 computer programming positions in a company, and 4 of the 9 people are men. If all 9 are qualified, in how many ways can the employer fill the computer programming positions if exact ...

In a normal distribution what percentage of scores fall

In a normal distribution, what percentage of scores fall between -1 and +1 standard deviations from the mean? Between the mean and -2 standard deviations ? Between the mean and +3 standard deviations?

The amount of soda a dispensing machine pours into a 12

The amount of soda a dispensing machine pours into a 12 ounce can of soda follows a normal distribution with a standard deviation of 0.08 ounce. Every can that has more than 12.20 ounces of soda poured into it causes a s ...

Statistics paperthe final paper provides you with an

STATISTICS Paper The Final Paper provides you with an opportunity to integrate and reflect on what you have learned during the class. The question to address is: "What have you learned about statistics?" In developing yo ...

Twenty years ago 40 children were born in the same hospital

Twenty years ago 40 children were born in the same hospital in Denmark. Researchers classified the children by whether or not they were breast-fed for at least 3 months. They were interested in whether breast-feeding ten ...

Business quantative analysis and statistics 1project

BUSINESS QUANTATIVE ANALYSIS AND STATISTICS 1 Project Management Scenario: You are a consultant with expertise in project management strategies. Select a possible project management situation from your current or future ...

1 why must a different t-test be used for matched-subjects

1. Why must a different t-test be used for matched-subjects and within-subjects designs than for randomized groups designs? 2. What was W. S. Gosset's contribution to behavioral research?

The potential revenues of all projects are in fact

The potential revenues of all projects are in fact uncertain. The company determines that the revenue for project 1 is following a uniform distribution ranging from $1,200,000 to $2,000,000. The revenue for project 2 is ...

Question 1 dr rasp just finished re-reading charles dickens

Question 1. Dr. Rasp just finished re-reading Charles Dickens' A Christmas Carol, a tragic novel in which a prosperous, hard-working businessman named Scrooge degenerates into a sentimental softie. This has prompted Dr. ...

Construct an aumann model of incomplete information for

Construct an Aumann model of incomplete information for each of the following situations, and specify a state of the world in that model which corresponds to the situation (there are several possible ways of including hi ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen