Question

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years.

a) What proportion of the plan recipients would receive payments beyond age 68?

b) Find the age at which payments have ceased for approximately 97.5 % of the plan participants.

Answer 1

Solution :

Given ,

mean = = 68

standard deviation = = 3.5

P(x >68 ) = 1 - P(x<68 )

= 1 - P[(x -) / < (68-68) /3.5 ]

= 1 - P(z <0 )

Using z table

= 1 - 0.5000

= 0.5

b.

Using standard normal table,

P(Z < z) = 97.5%

= P(Z < z) = 0.975  

= P(Z <1.96 ) = 0.975

z = 1.96 Using standard normal table,

Using z-score formula  

x= z * +

x= 1.96*3.5+68

x= 74.86