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# Thread: Please break the solution into more steps to help me understand

1. ## Please break the solution into more steps to help me understand

(SOA 153 #18)

Two instruments are used to measure the height, h, of a tower. The error made by the
less accurate instrument is normally distributed with mean 0 and standard deviation
0.0056h . The error made by the more accurate instrument is normally distributed with
mean 0 and standard deviation 0.0044h .
Assuming the two measurements are independent random variables, what is the
probability that their average value is within 0.005h of the height of the tower?

(A) 0.38
(B) 0.47
(C) 0.68
(D) 0.84
(E) 0.90

Solution to be broken down further:

Solution: D
Let X1 and X2 denote the measurement errors of the less and more accurate instruments,
respectively. If N(,) denotes a normal random variable with mean  and standard
deviation , then we are given X1 is N(0, 0.0056h), X2 is N(0, 0.0044h) and X1, X2 are
independent. It follows that Y =
2 2 2 2
1 2 is N (0, 0.0056 0.0044 )
2 4
X X h h
= N(0,
0.00356h) . Therefore, P[0.005h  Y  0.005h] = P[Y  0.005h] – P[Y  0.005h] =
P[Y  0.005h] – P[Y  0.005h]
= 2P[Y  0.005h] – 1 = 2P 0.005
0.00356
Z h
h
  

 1 = 2P[Z  1.4] – 1 = 2(0.9192) – 1 = 0.84.

2. http://www.soa.org/Files/Edu/edu-exam-p-sample-quest.pdf
http://www.soa.org/Files/Edu/edu-exam-p-sample-sol.pdf