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

Prove that if the result of the mens courtship algorithm

Prove that if the result of the men's courtship algorithm yields the same result as the women's courtship algorithm, then this resulting matching is the unique stable matching.

Assignmentstatistical analysisdirections be sure to save an

ASSIGNMENT Statistical Analysis Directions: Be sure to save an electronic copy of your answer before submitting it to Ashworth College for grading. Unless otherwise stated, answer in complete sentences, and be sure to us ...

A multiple-choice test consists of 7 questions each

A multiple-choice test consists of 7 questions. Each question has answer choices of a, b, c, and d, and only one of the choices is correct. If a student randomly guesses on each question, what is the probability that she ...

1nbspthe owner of a professional basketball team claims

1) The owner of a professional basketball team claims that the mean attendance at games is over 25,000 and therefore the team needs a new arena. Determine whether the hypothesis test for this claim is left-tailed, right- ...

Name and explain the contributions of three abolitionists

Name and explain the contributions of three Abolitionists that supported and helped African Americans before the Civil War fight slavery?

Jenks co partially refunded 180000 of its outstanding 10

Jenks Co. partially refunded $180,000 of its outstanding 10% note payable, made 1 year ago to Arma State Bank by paying $180,000 plus $18,000 interest (having obtained the $198,000 by using $52,400 cash and signing a new ...

1 why are psychologists sometimes interested in

1. Why are psychologists sometimes interested in epidemiology? 2. Why do researchers use probability rather than nonprobability samples when doing descriptive research?

According to a recent poll 29 of adults in a certain area

According to a recent poll, 29% of adults in a certain area have high levels of cholesterol. They report that such elevated levels "could be financially devastating to the regions healthcare system" and are a major conce ...

Problem 1somewhere in the milky way galaxy a class of 2000

Problem 1 Somewhere in the Milky Way Galaxy, a class of 2,000 students took a course in Astronomy.  The first exam scores and the final exam percentage reached earth, but transmission broke off after only a dozen student ...

Tell which measures are resistant to outliers and which are

Tell which measures are resistant to outliers and which are NOT resistant to outliers from the following list, and explain WHY or WHY NOT for each measure: Mean, median, Quartiles, correlation, and standard deviation.

  • 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