

2) The probability generating function for the random variable Zis Calculate the value of ρ (Z, 2Z)
2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1 2. 80 4. Prove that lim n-+v5n 0. -7 9 - in 5. Prove that lim n0 8 + 13n 13
Zis a standard nomal variable, find the probability PC -0.73 <2<227). Round to four decimal places. OA. 0.2211 OB. 0.7557 OC. O 4884 OD. 1.5400
2. Prove by induction that Ση.c)-(7+1) for n > 0 and i > 0.
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
2. Prove that lim (-1)"+1 = 0. n-00 n
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
how to prove this theorem?
Theorem 3. If e, 2 0,f, 2 0, (i-1, 2,. . . , k), and/pz' . . -P",; then k.
Surjection. Prove
F: A => 13 is surjective . YE B. Z = A - F (17.3) CA 1. Show : 9:Z=> 13 - {Yo], given g(x)= fx) for X6 Zis well-defined function 2. Shour: g is surjective
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...