Question

DE Suppose the IID random sample is (X,Y) where X, and Y, are independent random variables having Normal(11.1) and Normal(j2, 1) densities. So for X f( 1) = .7 exp f(y H1) = 7exp (yi - 12) For the following two free response questions, identify the density and the parameters of distributions of each quantity. X/01 – 2 n(x - H)2 + (n - 1)

Answer 1

Answer

given that

• $X_1,X_2,...,X_n\sim N(\mu_1,1)\Rightarrow \overline{X}\sim N(\mu_1,\frac{1}{n})\\ (using\ the\ additive\ property\ of\ normal\ distribution,i.e\ \sum_{i=1}^{n}X_i\sim N(n\mu_1,n))\\ similarly\ Y_1,Y_2,...,Y_n\sim N(\mu_2,1)\Rightarrow \overline{Y}\sim N(\mu_2,\frac{1}{n}) \\and\ \overline{X}\&\overline{Y}\ are\ independently\ distributed\\ So\ \frac{\overline{X}}{\sigma_1}\sim N\left ( \frac{\mu_1}{\sigma_1},\frac{1}{n\sigma_1^2} \right )\ \&\ 2\overline{Y}\sim N\left ( 2\mu_2,\frac{4}{n} \right )\\ \Rightarrow \frac{\overline{X}}{\sigma_1}-2\overline{Y}\sim N\left ( \frac{\mu_1}{\sigma_1}-2\mu_2,\frac{1}{n\sigma_1^2}+\frac{4}{n} \right )\\ Thus\ \frac{\overline{X}}{\sigma_1}-2\overline{Y}\ has\ normal\ density\ with\ mean=\frac{\mu_1}{\sigma_1}-2\mu_2\ and\\ variance=\frac{1}{n\sigma_1^2}+\frac{4}{n}$

$now\ (\overline{X}-\mu_1)\sim N(0,\frac{1}{n})\Rightarrow \sqrt{n}(\overline{X}-\mu_1)\sim N(0,1)\\ \Rightarrow n (\overline{X}-\mu_1)^2\sim \chi^2_1[\because X\sim N(0,1)\Rightarrow X^2\sim \chi^2_1]\\ and\ (n-1)S_Y^2\sim \chi^2_{n-1}\\ and\ n (\overline{X}-\mu_1)^2\&(n-1)S_Y^2\ are\ independently\ distributed\\ \Rightarrow n (\overline{X}-\mu_1)^2+(n-1)S_Y^2\sim\chi^2_{n}\\ Thus\ n (\overline{X}-\mu_1)^2+(n-1)S_Y^2\ has\ chi-square\ density\ with\\ degrees\ of\ freedom=n$