Question

The number of customers arriving per day at a certain automobile service facility is assumed to follow a Poisson Distribution with mean λ=20. Use normal approximation to find the probability that in a given day.

a) less than 25 customers will arrive;

b) at least 10 customers will arrive;

c) between 15 and 19 inclusive will arrive;

Answer 1

$Using\; normal\; approximation\;\\ \mu =\lambda=20, \sigma= \sqrt{\lambda}= \sqrt{20}\approx4.47\\ a)\;P(x< 25)=P(x<25.5)=P(z<\frac{25.5-20}{4.47})\\ =P(z<1.23)=0.5+P(0<z<1.23)\\ =0.5+0.3907=0.8907\\ b)\;P(x\geq 10)=P(x\geq 9.5)=P(z\geq \frac{9.5-20}{4.47})\\ =P(z\geq -2.35)=0.5+P(0<z<2.35)\\ =0.5+0.4906=0.9906\\ c)\;P(15<x< 19)=P(14.5<x<19.5)\\ =P(\frac{14.5-20}{4.47}<z<\frac{19.5-20}{4.47})\\ =P(-1.23<z<-0.11)\\ =P(0<z<1.23)-P(0<z<0.11)\\ =0.3907+0.0438=0.3469\\$