Chapter 7.2, Problem 23E

Applet Exercise

1. a Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y)] when Y has χ² distributions with 10, 40, and 80 df.
2. b What did you notice about P[Y > E(Y)] as the number of degrees of freedom increases as in part (a)?
3. c How does what you observed in part (b) relate to the shapes of the χ² 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 χ² distribution with ν df if and only if Y has a gamma distribution with α = ν/2 and β = 2.

1. a Use the applet Comparison of Gamma Density Functions to graph χ² densities with 10, 40, and 80 df.
2. b What do you notice about the shapes of these density functions? Which of them is most symmetric?
3. c In Exercise 7.97, you will show that for large values of ν, a χ² random variable has a distribution that can be approximated by a normal distribution with μ = ν and $\sigma =\sqrt{2v}$. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the χ² random variable Y?
4. d Refer to the graphs of the χ² densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the χ² distribution can be approximated with a normal distribution. Does this surprise you? Why?