2. Suppose that < a> is a cyclic group of order 10. Find all the generators in terms of a)
Show that if G is a group of order np where p is prime and 1 <n<p, then G is not simple.
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]
0 Let (f.) be a group, show that (ly) G where Gly); = Lael | ag= ga & gely is the center of G. (So, show that cly)< ; & cgjat. ) @ let y be a group, gel & Haf. Prove that Ks4 where us Ki Cig) := {acly I ag = gay is the centralizer of g inily, and K: N (H): = hatly I aH=Hay in the normalizen of Henly.
< 0) = 1/3, and Exercise 9.8. Suppose X has an N(u,02) distribution, P(X P(X < 1) = 2/3. What are the values of u and o?!
Provided N(0, 1) and without using the LSND program, find P( - 2 <3 <0) Provided N(0, 1) and without using the LSND program, find P(Z < 2). Provided N(0, 1) and without using the LSND program, find P(Z <OOR Z > 2). Message instructor about this question Provided N(0, 1) and without using the LSND program, find P(-1<2<3). 0.84 Message instructor about this question
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.
fx (z)='0 otherwise Let Xa)<...<Xn) be the order statistics. Show that Xa)/X(n) and X(n) are independent random variables.
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
In the problems below, give the order of the element in the indicated factor group. (a) in (b) in (c) in (5)(20 points) In the problems below, give the order of the element in the indicated factor group. (a) (1, 2)+ < (1,1) > in Z3 x Z6/ < (1,1) >. (b) (3, 2)+ < (4,4) > in Z6 * Z8/ < (4,4) >. (c) 26+ < 12 > in Z60/ <12>.