Question

An insurance company has 10,000 automobile policy holders. The expected yearly claim per policyholder is $240, with a standard deviation of $800. Approximate the probability that the total yearly claims:

a) exceeds $2 million

b) is less than $1.5 million

Answer 1

X ~ N ( n , n} ) = ( 10000 * 240 , 10000 * 8002 ) = ( 2400000 , 80000)

= N ( 2.4 , 0.8) in millions

a)

X ~ N ( µ = 2.4 , σ = 0.8 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )   
P ( X > 2 ) = P(Z > (2 - 2.4 ) / 0.8 )   
= P ( Z > -0.5 )   
= 1 - P ( Z < -0.5 )   
= 1 - 0.3085

= 0.6915
  

b)

X ~ N ( µ = 2.4 , σ = 0.8 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 1.5 ) = P ( Z < 1.5 - 2.4 ) / 0.8 )
= P ( Z < -1.13 )
P ( X < 1.5 ) = 0.1292