Question

The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter

μ = 40.

(Round your answers to four decimal places.)

(a)

Calculate the approximate probability that between 35 and 70 tickets are given out on a particular day.

(b)

Calculate the approximate probability that the total number of tickets given out during a 5-day week is between 195 and 265.

(c)

Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations.

Calculate the exact probability that between 35 and 70 tickets are given out on a particular day.

also Calculate the exact probability that the total number of tickets given out during a 5-day week is between 195 and 265.

Answer 1

SOLUTION:

From given data,

The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter μ = 40. (Round your answers to four decimal places.)

Using

Normal approximation to Poisson distribution.

μ = 40.

= 40

= sqrt() = sqrt(40) = 6.32455

z = (x-μ ) / = (x-40) / 6.32455

(a) Calculate the approximate probability that between 35 and 70 tickets are given out on a

particular day.

The approximate probability that between 35 and 70 tickets are

P(35 < x < 70 ) = P((35-40) / 6.32455​​​​​​​ < (x-μ ) / < (70-40) / 6.32455​​​​​​​)

P(35 < x < 70 ) = P(-0.79 ​​​​​​​ < z < 4.74)

P(35 < x < 70 ) = P( z < 4.74) - P( z < -0.79)

P(35 < x < 70 ) = 1 - 0.214764

P(35 < x < 70 ) = 0.785236

(b) Calculate the approximate probability that the total number of tickets given out during a 5-day week is between 195 and 265.

The approximate probability that between 195 and 265 tickets are

Where,

μ = 40*5 = 200

= sqrt(*5) = sqrt(40*5) = sqrt(200) = 14.1421356

P(195 < x < 265 ) = P((195-200) / 14.1421356​​​​​​​ < (x-μ ) / < (265-200) / 14.1421356​​​​​​​)

P(195 < x < 265 ) = P(-0.35 ​​​​​​​ < z < 4.59)

P(195 < x < 265 ) = P( z < 4.59) - P( z < -0.35)

P(195 < x < 265 ) = 1 - 0.363169

P(195 < x < 265 ) = 0.636831

(c) Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations.

Calculate the exact probability that between 35 and 70 tickets are given out on a particular day.

= > P( z < 4.74) - P( z < -0.79)

=> 0.785236

also Calculate the exact probability that the total number of tickets given out during a 5-day week is between 195 and 265.

=>P( z < 4.59) - P( z < -0.35)

=> 0.636831