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Applet Exercise As we stated in Definition 4.10, a random variable Y has a x² distribution with v df if and only if Y has a gamma distribution with a = v/2 and ß = 2. a Use the applet Comparison of Gamma Density Functions to graph x² densities with 10, 40, and 80 df. b What do you notice about the shapes of these density functions? Which of them is most symmetric? c In Exercise 7.97, you will show that for large values of v, a x² random variable has a distribution that can be approximated by a normal distribution with μ = vand o = √2v. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the x² random variable Y? d Refer to the graphs of the x² densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the x² distribution can be approximated with a normal distribution. Does this surprise you? Why?

Applet Exercise As we stated in Definition 4.10, a random variable Y has a x² distribution with v df if and only if Y has a gamma distribution with a = v/2 and ß = 2. a Use the applet Comparison of Gamma Density Functions to graph x² densities with 10, 40, and 80 df. b What do you notice about the shapes of these density functions? Which of them is most symmetric? c In Exercise 7.97, you will show that for large values of v, a x² random variable has a distribution that can be approximated by a normal distribution with μ = vand o = √2v. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the x² random variable Y? d Refer to the graphs of the x² densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the x² distribution can be approximated with a normal distribution. Does this surprise you? Why?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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7.23 Applet Exercise a Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y)] when Y has x² distributions with 10, 40, and 80 df. b What did you notice about P[Y> E(Y)] as the number of degrees of freedom increases as in part (a)? c How does what you observed in part (b) relate to the shapes of the x² densities that you obtained in Exercise 7.22?
Transcribed Image Text:7.23 Applet Exercise a Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y)] when Y has x² distributions with 10, 40, and 80 df. b What did you notice about P[Y> E(Y)] as the number of degrees of freedom increases as in part (a)? c How does what you observed in part (b) relate to the shapes of the x² densities that you obtained in Exercise 7.22?
7.22 Applet Exercise As we stated in Definition 4.10, a random variable Y has a x² distribution with v df if and only if Y has a gamma distribution with a = v/2 and ß = 2. a Use the applet Comparison of Gamma Density Functions to graph x² densities with 10, 40, and 80 df. b What do you notice about the shapes of these density functions? Which of them is most symmetric? c In Exercise 7.97, you will show that for large values of v, a x² random variable has a distribution that can be approximated by a normal distribution with μ = vand o = √2v. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the x² random variable Y? d Refer to the graphs of the x² densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the x² distribution can be approximated with a normal distribution. Does this surprise you? Why?
Transcribed Image Text:7.22 Applet Exercise As we stated in Definition 4.10, a random variable Y has a x² distribution with v df if and only if Y has a gamma distribution with a = v/2 and ß = 2. a Use the applet Comparison of Gamma Density Functions to graph x² densities with 10, 40, and 80 df. b What do you notice about the shapes of these density functions? Which of them is most symmetric? c In Exercise 7.97, you will show that for large values of v, a x² random variable has a distribution that can be approximated by a normal distribution with μ = vand o = √2v. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the x² random variable Y? d Refer to the graphs of the x² densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the x² distribution can be approximated with a normal distribution. Does this surprise you? Why?
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