Question

Suppose that x has a Poisson distribution with μ = 22.

(a)	Compute the mean, μx, variance, σ2xσx2, and standard deviation, σx. (Do not round your intermediate calculation. Round your final answer to 3 decimal places.)

µx = , σx2 = , σx =

(b)

Calculate the intervals [μx ± 2σx] and [μx ± 3σx ]. Find the probability that x will be inside each of these intervals. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to the previous whole number. (Use the value for standard deviation from part a. Round all your answers to 4 decimal places. Negative amounts should be indicated by a minus sign.)

[µx ± 2σx]	[ , ]
P( ≤x ≤ )
[µx ± 3σx]	[ , ]
P( ≤x ≤ )

Answer 1

a) = 22

b) = 22 - 2 * 4.69 = 12.62 = 13

= 22 + 2 * 4.69 = 31.38 = 31

= 22 - 3 * 4.69 = 7.93 = 8

= 22 + 3 * 4.69 = 36.07 = 36

P(13 < X < 31) = P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) + P(X = 21) + P(X = 22) + P(X = 23) + P(X = 24) + P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28) + P(X = 29) + P(X = 30) + P(X = 31)

= e^(-22) * ((22)^13/13! + (22)^14/14! + (22)^15/15! + (22)^16/16! + (22)^17/17! + (22)^18/18! + (22)^19/19! + (22)^20/20! + (22)^21/21! + (22)^22/22! + (22)^23/23! + (22)^24/24! + (22)^25/25! + (22)^26/26! + (22)^27/27! + (22)^28/28! + (22)^29/29! + (22)^30/30! + (22)^31/31) = 0.9584

P(8 < X < 36) = P(X = 8)+ P(X = 9)+ P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) + P(X = 21) + P(X = 22) + P(X = 23) + P(X = 24) + P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28) + P(X = 29) + P(X = 30) + P(X = 31) + P(X = 32) + P(X = 33) + P(X = 34) + P(X = 35) + P(X = 36)

= e^(-22) * ((22)^8/8! + (22)^9/9! + (22)^10/10! + (22)^11/11! + (22)^12/12! + (22)^13/13! + (22)^14/14! + (22)^15/15! + (22)^16/16! + (22)^17/17! + (22)^18/18! + (22)^19/19! + (22)^20/20! + (22)^21/21! + (22)^22/22! + (22)^23/23! + (22)^24/24! + (22)^25/25! + (22)^26/26! + (22)^27/27! + (22)^28/28! + (22)^29/29! + (22)^30/30! + (22)^31/31 + (22)^32/32! + (22)^33/33! + (22)^34/34! + (22)^35/35! + (22)^36/36!) = 0.9976