Question

Please do both (a) and (b) and fully explain in detail.

Problem 4. Chernoff bound for a Poisson random variable. Let X be a Poisson random variable with parameter λ (a) Show that for every s 20, we have (b) Assuming that k > λ, show that

Answer 1

I Page 6 Cheaa Page 6 WANASOL 5.062 now here M = Mean of x. For y Poi (A) Elx) = : P[X> (148) 17 1 (17) CH4

I Page 7 Page 7 Chegal 96) Now here k= or K (1+8). 48 and 8 = i sabetuhing we have.

Page 8 Cheag I Pages 74 le & e & + t. & X.

9 Page Cheap I legea 9 - 05) PCX7 Le Kad а)

Page 10 tcheon ГРес BO) . Contoh - PX>k] Le tekak Ekk. a P [X Zk] < é dedor fence Prored

1 Page 1 Cheral I Page 1 ô (4) The Chenott bound B de tined as follows: D ezcally. PL1X76] SE[ex] - elak a P [X*] é Ele ex] e ek

- Page 2 Chega I Page 2 PL x = a - en La 2 OL 2,. . Now E lexy

| Page 3 | CHESS | PAGES 1 sca) e 82 e-da e exa - to 08 SUNDAY e les

I Page 4 [ chop Page 41 SC) P [X>k] = è dete P[XZk] s elete- Hence Prored.

Page 5 | Cred e coill we the another form of Chernote boud which is as follows: PEX7 (148) u] IM La L S es Lato (HO)