Question

We toss a fair coin n 400 times and denote Zn the number of heads. (a) What are E(Zn) et Var(Zn)1? (b) What is the probability that Z 200? (use the normal approximation together with the continuity correction (c) What is the smallest integer m such that Pr 200-mくZ.く200 +m] > 20%? (use the normal approximation together with the continuity correction).

Answer 1

Zn denores he numb en of head s Total navn bev, of hosses is n: 400 enoeS eting head As the coin is faiv,so 2 we assume hat the sses are independenh Sn, Zn ~ Bin ( n: 400, P: 0.5丿 40 ox6.5 2 00 Var (2)nplie 100 We kn b w then E(x)np and

b we know that if arandem vavn a bl e X ~ Bin(n, p) and n i s sufficiently Large So that beth Nov, 200, 10 0 aA nN I D Zn- 2o 0 199.5-20 0 1 0 1 0

·X-200 e n then X-N1) -0. 0ら (w) is e edf of stand and 2 х ф(0.05)-1 2x 0 52 3 I O0 462 c Hen 0,20 n 200 m)- plZn 4 200m 7 20

o, Zn 200 + mt 0.5 -P2n 200 - m-0570 20 usinaconhinui Zn-20o 1 0 Zn 200199 5 -m 200 1 0 1 0 0 20 m 0 5 D 20 m+0.5 1 0 1 0 0.20 mt0 5 1 o 0.20 05

mt 0.5 OV, m0.5 0 25 33 0 V a z vn 10x0. 2533-0.5 OV m2 033. m2 2 033 andm is an integer So, the smallest inhegev, İs