A department store has 3 doors. Arrivals at each door form Poisson processes wit

Question

A department store has 3 doors. Arrivals at each door form Poisson
processes with rates ‚1 = 110, ‚2 = 90, ‚3 = 160 customers per hour.
30% of all customers are male. The probability that a male customer
buys something is 0:80, and the probability of a female customer buying
something is 0:10. An average purchase is 4:50 euros

a) What is the average worth of total sales made in a 10 hour day?
b) What is the probability that the third female customer to purchase
anything arrives during the first 15 minutes? What is the expected
time of her arrival?

Explanation / Answer

A(a) Here Pr(Male) = 0.30 ; Pr(Female) = 0.70

Pr(Male customer buy something) = 0.80 ; Pr(female buy something) = 0.10

Pr(Any customer buy something) = 0.80 * 0.30 + 0.70 * 0.10 = 0.31

Average ourchase value = 4.50 euros

average Total customers arrival in one hour = 110 + 90 +160 = 360

Average worth of total sales made in a 10 hour day = 360 * 10 * 4.50 * 0.31 = 5022 euros

(b) Here we have to find the probability that the third female customer to purchase anything arrives during the first 15 minutes. So, we have to find first that at most there is 2 female customer to purcahse anything in the first 15 minutes.

expected number of female customer to purchase anything arrives during the first 15 minutes = 360 * 0.25 * 0.70 * 0.10 = 6.3

so If X is the number of female customers who purchase anything in 15 minutes

so,

Pr(X <= 2; 6.3) = 0.0498

so Pr(Third female customer to purchase anything arrives during the first 15 minutes) = 1- 0.0498 = 0.9502

Expected time of her arrival = (15/6.3) * 3 = 7.15 minutes

so the third female customer to purchase anything arrives have an expected time of 7.15 minutes

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