Hi Swati,
Here, mu(49) = Theta(49) / E49c = 42 / E49c
We have to make the denominator correspond to the numerator. This is the PRINCIPLE OF CORRESPONDENCE.
Here, theta*(49) = number of DEATHS aged 49 nearest BDay
I invent P* so that it corresponds with the definition of Theta(49):
P*(49,t) = number of PEOPLE aged 49 nearest BDay at time t
This bit is easy, you just copy and write "people" instead of "deaths".
Now, E49c = integral t = 0 to 2 of P*(49,t)
which we approximate using trapezium rule as 0.5 P*(49,0) + P*(49,1) + 0.5 P*(49,2)
All that remains is to approximate P*(49,t) (the data we want) from P(49,t) (the data we have)
P*(49,t) = number of PEOPLE aged 49 nearest BDay
P(x,t) = number of PEOPLE aged 49 last BDay
The people aged 49 nearest birthday will be between the ages of 48.5 and 49.5. So, half of them are 48 on their last birthday and half of them are 49 on their last birthday:
P*(49,t) = 0.5 {P(48,t) + P(49,t)}
Copy this in for t = 0,1 2 above:
E49c = 0.5*0.5{P(48,0) + P(49,0)} + 0.5{P(48,1) + P(49,1)}
+ 0.5*0.5 {P(48,2) + P(49,2)}
= 0.5*0.5{3486 + 3450} + 0.5{3384 + 3507}
+ 0.5*0.5 { 3420 + 3435}
= 6893.25
So, mu(49) = 42 / 6893.25 = 0.0060929
Check my numbers but I hope this is the answer.
Good luck!
John
Last edited: Aug 20, 2014
