Suppose that 100 airlines organize to determine the likeliho

Suppose that 100 airlines organize to determine the likelihood for an individual airline to have a plane crash. Let X_1, X_2 ..., X _100 represent the number of crashes for each airline: that is, X_i is the number of crashes that airline i has. The organization found that the X_i are independent (might not be a reasonable assumption, but let\'s go with it) and identically distributed as Poisson random variables w ith parameter pie = 0.4. Now let X be the number of crashes lor all of the airlines. First calculate exactly P(X

Solution

a).

=0.4

X = sum of 100 random variables with =100*0.4 = 40

P(X=x) = e^{- x} / x!

P( x <45) = 0.7657

b).

X follows normal with N( 40, sqrt(40)) = N( 40,6.3246)

Z value for 45, z = (45- 40)/6.3246 =0.79

P( X<45) = P( z <0.79) = 0.7852