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Chapter 12: Q12.3-2E (page 815)

Describe how to convert a random sample \({{\bf{U}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{U}}_{\bf{n}}}\) from the uniform distribution on the interval \({\bf{[0,1]}}\) to a random sample of size \({\bf{n}}\) from the uniform distribution on the interval\({\bf{[a,b]}}\).

Short Answer

Expert verified

The Variable is \({V_i} = a + (b - a){U_i}.\)

Step by step solution

01

Definition for uniform random sample:

The samples are distributed uniformly across the half-open interval [low, high] (includes low, but excludes high). In other words, uniform is equally likely to draw any value within the given interval.

02

Find a random variable from uniform distribution:

The\(cdf\)of a random variable from uniform distribution on interval\([a,b]\)is

\(F(x) = \frac{{x - a}}{{b - a}},\;\;\;a \le x \le b\)

Zero for\(x < a\)and\(1\)for\(x > b\)

Define random variables\({V_i},i = 1,2, \ldots ,n\)as

\({V_i} = a + (b - a){U_i},\;\;\;i = 1,2, \ldots ,n,\)

Where\({U_i}\)are as defined in the exercise.

Obviously, such random variable has uniform distribution with pdf

\(f(v) = \frac{1}{{b - a}},\;\;\;a \le v \le b\)

Or equally, it has uniform distribution on interval\([a,b]\).

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Most popular questions from this chapter

Use the beef hot dog data in Exercise 7 of Sec. 8.5. Form 10,000 nonparametric bootstrap samples to solve this exercise.

a. Approximate a 90 percent percentile bootstrap confidence interval for the median calorie count in beef hot dogs.

b. Approximate a 90 percent percentile-t bootstrap confidence interval for the median calorie count in beef hot dogs.

c. Compare these intervals to the 90 percent interval formed using the assumption that the data came from a normal distribution.

Test the standard normal pseudo-random number generator on your computer by generating a sample of size 10,000 and drawing a normal quantile plot. How straight does the plot appear to be?

Let \({x_1},...,{x_n}\) be the observed values of a random sample \(X = \left( {{x_1},...,{x_n}} \right)\) . Let \({F_n}\)be the sample c.d.f. Let \(\,{j_1},...\,,{j_n}\) be a random sample with replacement from the numbers \(\left\{ {1,.....,n} \right\}\) Define\({x_i}^ * = x{j_i}\) for \(i = 1,..,n.\)ashow that \({x^ * } = \left( {{x_1}^ * ,...,{x_n}^ * } \right)\) is an i.i.d. sample from the distribution\({F_n}\)

\({{\bf{x}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}.....{{\bf{x}}_{\scriptstyle{\bf{n}}\atop\scriptstyle\,}}\) be uncorrelated, each with variance \({\sigma ^2}\) Let \({{\bf{y}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}.....{{\bf{y}}_{\scriptstyle{\bf{n}}\atop\scriptstyle\,}}\) be positively correlated. each with variance, prove that the variance of \(\overline x \)is smaller than the variance of \(\overline y \)

Use the data consisting of 30 lactic acid concentrations in cheese,10 from example 8.5.4 and 20 from Exercise 16 in sec.8,6, Fit the same model used in Example 8.6.2 with the same prior distribution, but this time use the Gibbs sampling algorithm in Example 12.5.1. simulate 10,000 pairs of \(\left( {{\bf{\mu ,\tau }}} \right)\) parameters. Estimate the posterior mean of \({\left( {\sqrt {{\bf{\tau \mu }}} } \right)^{ - {\bf{1}}}}\), and compute the standard simulation error of the estimator.
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