Assuming the number of white blood cells per unit of volume

Assuming the number of white blood cells per unit of volume diluted blood counted under a microscope follows a Poisson distribution with a mean =100, what is the probability, using a normal approximation, that a count of 90 or less will be observed?

Solution

let X be the random variable denoting the number of white blood cells per unit of volume diluted blood counted under a microscope.

given that X follows a Poisson distribution with a mean =100.

so E[X]=100 and variance is V[X]=100

hence standard deviation would be SD(x)=sqrt[V(X)]=sqrt(100)=10

so under normal approximation we have (X-100)/10 follows approximately a N(0,1) distribution.

now required probability

P[X<=90]=P[(X-100)/10<=(90-100)/10]=P[Z<=-1] where Z~N(0,1)

             =0.158655   [answer]      [using MINITAB]