Let Z be a standard normal random variable and let V have a chi-square distribution with n-degrees of freedom. Assume that Z and V are independent and let

T = Z / √ (V/n)

Find the density of T (The distribution of T is known as the t-distribution with n degrees of freedom.)

Solution Preview

Please refer to the attachment.

A statistical distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose "Student." Given n independent measurements , let
(1)
where is the population mean, is the sample mean, and s is the estimator for population standard deviation (i.e., s=SQRT(V)) defined by (2)
Student's t-distribution is defined as the distribution of the random variable t which is (very loosely) the "best" ...

Solution Summary

Let Z be a standard normal random variable and let V have a chi-square distribution with n-degrees of freedom. Assume that Z and V are independent and let

Find the density of T (The distribution of T is known as the t-distribution with n degrees of freedom.)

· Calculate probabilities and jointprobabilities on the events of driving to work everyday for 10 days.
· Identify the meaning of independent and dependent of this event
· Understand the basic logic of probability theory
Monday 30 minutes
Tuesday 35 minutes
Wednesday 30 minutes
Thursday 25 minute

I have been struggling with theses two following problems:
illustration: age a(0.00%) b(0.01-0.9%) c(>_0.10%)
d 0-19 142 7 6 155
e 20-39 47 8 41 96
f 40-59 29 8 77

(See attached file for full problem description with equations)
---
53. Given that the joint pdf of the random variables X and Y is defined by
(i) Find the number k.
(ii) Find the marginal pdf's fX and fY. Are X and Y independent?
(iii) Calculate the probabilities:
---

Let X1 and X2 be two independent standard normal random variables. Let Y1 = X1+X2 and Y2=X1/X2. a) Find the joint density of Y1 and Y2 b) Find the marginal density of Y1 and Y2 (The distribution of Y2 is known as the Cauchy distribution).

A retail outlet receives radios from three electrical appliance companies. The outlet receives 20% of its radios from A, 40% from B, and 40% from C. The probability of receiving a defective radio from A is .01; from .02; and from C .08.
A. Develop a probability tree showing all marginal, conditional and jointprobabilities.

A metropolitan school system consists of three school districts-norths, south, central. The north district contains 25% of all students, the south district contains 40%, and the central district contains 35%. A minimum-competency test was given to all students; 10% of the north district students failed, 15% of the south distri

Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as tabled.
Income = $40,000 Income > $40,000
Accept offer 40 30
Reject offer 210

You have a standard deck of cards and you take out 2 cards at random. Let Y1 represent the number of red Queens in your sample and let Y2 represent the number of spades in your sample.
a) calculate p. it is mean p(Y1, Y2);
b) calculate var (Y1I Y2=1)
c) Determine E (Y2I Y1=0)

Each salesperson at stiles compton is rated either below average, average, or above average with respect to sales ability. Each salesperson is also rated with respect to his or her potential for advancement: either fair, good or excellent. These traits for the 500 sales people were cross classified into the following table.

A metropolitan school system consists of three school districts-norths, south, central. The north district contains 25% of all students, the south district contains 40%, and the central district contains 35%. A minimum-competency test was given to all students; 10
% of the north district students failed, 15% of the south dis