Applet Exercise As we stated in Definition 4.10, a random

Chapter 7, Problem 22E

(choose chapter or problem)

Applet Exercise As we stated in Definition 4.10, a random variable Y has a \(x^{2}\) distribution with ν df if and only if Y has a gamma distribution with \(\alpha=v / 2 \text { and } \beta=2\)

Use the applet Comparison of Gamma Density Functions to graph \(x^{2}\) densities with 10, 40, and 80 df.What do you notice about the shapes of these density functions? Which of them is most symmetric?In Exercise 7.97, you will show that for large values of ν, a \(x^{2}\) random variable has a distribution that can be approximated by a normal distribution with \(\mu=v \text { and } \sigma=\sqrt{2 v}\) How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the \(x^{2}\) random variable Y?Refer to the graphs of the \(x^{2}\) densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the \(x^{2}\) distribution can be approximated with a normal distribution. Does this surprise you? Why?

Equation Transcription:

   

   

   

 

 

 

Text Transcription:

x^2    

\alpha =v/2  and \beta =2

x^2  

x^2  

\mu =v and \sigma =√2v

x^2

x^2

x^2