Question

We will need this important result of this problem for something that is coming up in class! Suppose that X1, X2, ..., Xn 10 N(4, 02). (a) Show that 2-1(Xi – u)2 -1(X; – X) n(X – u)2 02 o2 T 02 (This has nothing to do with the particular distribution here.) (b) Write down the joint pdf for X1, X2, ..., Xn and use the above to rewrite the "e-exponent part”. (c) Consider the joint transformation Y1 = X, Y; = Xi – X for i = 2,3,..., n. Use the Jacobian method to find the joint pdf for Yı, Y2, ..., Yn. Show that Y¡ is inde- pendent of Y2, ..., Yn. Note: Just generalize the bivariate transformation. Also, you might want to take advantage of the following two facts. 1) The determinant of a diagonal matrix (all entries off of the diagonal are zero) is the product of the diagonal entries. 2) If you take any row of a matrix, multiply it by a scalar, and add it to another row, the determinant of the matrix will not change. (d) Show that X1 - X = -21-2(Xi – X). Conclude that X1 - X is independent of X. (e) Conclude that X is independent of the sample variance S2! Wow!

Answer 1

→ Suppose that X1, X 29..., Xn id N(u,0-2) - 02 02 L.H.S, i= 1 (@) We hove to show that, 21. (x; -m) 27., (x-7) n (8 - x)? 02 L.H.S, (xi - we 02 -3 *-***-w)] { x=>+<*-433] [{(x-x) + (-1) 22 (xi-x) (3 P) 2640" = 23c4->) 68-6)}] [ <=}+n67.*26-05] (3:=)] ar 2 . (X-M) 21 (xi- 02 IME "Ms i = 1 9=1 [:( - ) is constant

(xi-x) = 0 = ) + n (x-u) 1 (x;- o2 Li=1 i=1 4136=>«(3-2)=0 *. 261-**=0] ² (xi-x) = n(X-42 m - RoHS orovede) 02 02

(6) The joint pdf of X19X2.. Xn is given by 1 - 2 E (xi-x)2 202 [(xi-u) 2 (der) none (xi-x) r: (8-4) 2 + - (dernon. & 2 1 1 S2(x-7)27 {n(x-M) 2 2 L2 (72) (120) :o (c) Let us consider the point transformation Y = X and and I = xi-x for i=2,6... a That is, (x2 X 29. 9Xn) (Y1oY2.... I on) such that, b.. / x1 72 .. . (-)/ → Y = P., Here, Pis an orthogonal matrix. Now, Y'I - X'P'PX = X'X n eie, S ee, T7.2 -7 X 2 i=1 i=1 The iaceobian of the transformation is Job - Det (p-1) = +1 2. lal=1

So, the joint pdf of Ya, Y2... In is given by 40 c J 1 ena sn (7-4)²) 02 (12) o 22 Y - 1 € 202 1 1 S G1-4)2 2 é 2021 e 202 1 - 2 1 825 tario veroo ... ker.o Toro = f(Y₂)... f(yn). f (Y) So, Y1 and Y,,..., Yn are independently distributede. (d) = xi/n i= 1 nx = x+ X2+...+Xn **** ********* = x - 3x: => X1 + Xi = X i=2 x2 i=2 = (x1 - x) = n*- = (1-1) = i=2 x - x ²x: i=2 = – ² (xi-x) .X2 - x = - Z (x; -7) (proved) 12 We know, X=Y 1 It can be seen that x2 -=- £ (2-) = -Y Since , Y,,....Yn and Y2 are independent Then, X and (x-) is also independent.

002 n- 1 i=1 cer (0) S c => =>(-1) 5º = (x+3) = (4-8247 (xi-x) = 2 (3-) + 2x: i=1 i = 2 n 22 {+ - 7,2 I i=2 ) ;=2 92 is dependent on Y2..... Yn and Thus, X and s2 is also independent. is dependent on Y.