Concept explainers
A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution
where α, β > 0.
a Find the density function.
b For fixed values of β and α, find a transformation G(U) so that G(U) has a distribution function of F when U has a uniform distribution on the interval (0, 1).
c Given that a random sample of size 5 from a uniform distribution on the interval (0, 1) yielded the values .0058, .2048, .7692, .2475 and .6078, use the transformation derived in part (b) to give values associated with a random variable with a Pareto distribution with α = 2, β = 3.
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Chapter 6 Solutions
Mathematical Statistics with Applications
- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.arrow_forwardFor the following table of data. x 1 2 3 4 5 6 7 8 9 10 y 0 0.5 1 2 2.5 3 3 4 4.5 5 a. draw a scatterplot. b. calculate the correlation coefficient. c. calculate the least squares line and graph it on the scatterplot. d. predict the y value when x is 11.arrow_forwardbThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.arrow_forward
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