Exam ST question (Poisson process)

Hi guys,



I'm having a problem in understanding the solution for one of ASM manual question(28.18).

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Starting at the time a supermarket opens, you are given:

- Women arrive in a Poisson process at a rate of 300 per hour, and leave in a Poisson process at a rate of 160 per hour.

- Men arrive in a Poisson process at a rate of 100 per hour, and leave in a Poisson process at a rate of 80 per hour.



Using the normal approximation, estimate the probability of more than 410 people in the store after 2.5 hours.

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I got E(2.5*T) = 2.5*(300+100-160-80) = 400, which is the correct mean

and I got Var(2.5*T) = (2.5)^2*(300+100+160+80).



However, the solution says, the correct variance to use for the normal approximation is 2.5*Var(T).



Can anyone tell me why 2.5*Var(T) is correct, rather than Var(2.5*T)?



Thank you

Exam ST question (Poisson process)