Q4. The number of items handed in to the Lost Property Office of a busy railway

Question

Q4. The number of items handed in to the Lost Property Office of a busy railway station each day is a Poi(15) random variable on a weekday (Monday to Friday), a Poi(22) random variable on a Saturday and a Poi(12) random variable on a Sunday. The numbers of items handed in to the office on different days are independent. Use a Normal approximation to find the approximate probability that no more than 100 items in total are handed in to this Lost Property Office next week (Sunday to Saturday, inclusive): Enter your answers as decimal numbers (4 decimal places are sufficient). The required probability is (3 marks)

Explanation / Answer

Ans:

As,the number of items handed into office on different days are independent.

So,equivalent mean rate for poisson distribution for all days=15*5+22+12=109

For Normal approximation:

mean=109

std dev=sqrt(109)=10.44

Now,100 in a week

z=(100-109)/10.44=-0.862

P(z<=-0.862)=0.1943

(using poisson distribution:P(x<=100)=POISSON(100,109,TRUE)=0.2093)

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