Question

26./ Let Xi, X2, X100 be i.i.d. Poisson (5) distributed. a./ Find the probability that Sioo is bigger than 550. b./ Find the probability that S100 is bigger than 5.2

Answer 1

a. X1, X2,....,X100 ~ Poi(5).

E(Xi) = 5 and Var(Xi) = 5.

Since, Xi's are i.i.d., thus,

E($S_{100}$) = 100 * 5 = 500 and Var($S_{100}$) = $(100)^2$ * 5 = 50000

Since, we have sufficiently large number of samples,

We can assume, $S_{100}$ ~ N(500, 50000)

i.e. Z = ($S_{100}$ - 500)/223.6068 ~ N(0,1).

Hence, the required probability = P($S_{100}$ > 550)

= P[($S_{100}$ - 550)/223.6068 > (550 - 500)/223.6068]

= 1 - P(Z < 0.2236) = 1 - $\Phi$ (0.2236) = 1 - 0.5885 = 0.4115. (Ans).

[$\Phi$(.) is the cdf of N(0,1)].

b. We have, E($\overline{S_{100}}$) = 5, Var($\overline{S_{100}}$) = 5/100 = 0.05,

s.d.($\overline{S_{100}}$) = 0.2236

Assuming Normal distribution, Z = ($\overline{S_{100}}$ - 5)/0.2236 ~ N(0,1)

P($\overline{S_{100}}$ > 5.2) = P[($\overline{S_{100}}$ - 5)/0.2236 > (5.2 - 5)/0.2236]

= 1 - P(Z < 2.2361) = 1 - 0.9873 = 0.0127. (Ans).