Question

(a) Let Xl have etti-quered distribution with parameter 1, and let Xz be independent of Xi and have a chi-squared distribution with parameter vs. Show that X1 + X2 ha chi-squared distribution with perampeter v 1 + 2 Let Y - X1 + X2. Identify the correct expression for Fyl- 272-, -+ *>72 berland TW) Fly) -=6{c(770705 044-1673) (70). b 6{"(70X70. --*|(2713).(770). 674-0).* 6(7-7). orsan = $(307+972 () 1.17) F4n = 6" (+0312 (23 * * *2) -6(***317.65 -)) 8*(****)?:(,>>).**. 1-2) Fan- Simplifying Fyr results in which of the following? 1/2 - + 1/2 - 1 Fyr - ( 23 Thus we see the integrand is the density chi-squared distribution with parerieter vi + v2 (b) It can be shown that if Z is a standard normal then 2 has a chi-squared distribution with v = 1. Let 21:22 - 2 ben independent standarc normelry's. What is the distribution of 2.2+ Oy () Z?+2 -Soed distribution with --Select-- se<z-23)+23has a Select distribution with Sole Select +2, Justify your answer Therefore, z? +...+2, has a select-- Cistribution with tel Let X. X., be a random sample from a normal distribution with many and varlance 02. What is the distribution of the sum Y - - * "? Justy vour answer. has a Select distribution with Sect which means the sun has a Select distribution with The cuantry* - has a Selet cistabudion with Selec -Select

Answer 1

2 b) z + 2² has a chia squared distribution with 12 degrees of freedom so (2,42%)+ Zz chi-squared / dismibution with 3 degrees of freedom 2 has a there for 2,4224... + 2 has a Ichisqueosed dishbuchen with n-degrees of freedom c) The quantity X3 - " has a Normal distribution with mean o and varlance 2 So Ichi-squared ( *--Me has a distribuhon with Idegrees of freedor which means the sum has a chi-squared distribuhon with degrees of freedom n