Two instruments are used to measure the height, [math]h[/math], of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056 [math]h [/math]. The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 0.0044 [math]h[/math]. The errors from the two instruments are independent of each other.

Calculate the probability that the average value of the two measurements is within 0.005 [math]h[/math] of the height of the tower.

• 0.38
• 0.47
• 0.68
• 0.84
• 0.90

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let T denote the number of days that elapse before a high-risk driver is involved in an accident. Then T is exponentially distributed with unknown parameter λ . Now we are given that

[[math]] 0.3 = P[T ≤ 50] =50 \int_0^{50}\lambda e^{-\lambda t} \, dt = 1 - e^{-50\lambda}. [[/math]]

Therefore, [math]e^{–50\lambda} = 0.7 [/math] or [math]\lambda = − (1/50) \ln(0.7) [/math]. It follows that

[[math]] P[T ≤ 80] = \int_0^{80} \lambda e^{-\lambda t} \, dt = 1 - e^{-80 \lambda} = 1- e^{(80/50) \ln(0.7)} = 1-(0.7)^{80/50} = 0.435. [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.