IAI Actuarial Core Principles · Actuarial Statistics · Basic univariate distributions and generating samples
A random variable X has cumulative distribution function F(x) = 1 - e^(-0.05x) for x > 0. Using the inverse transform method, a Uniform(0,1) value u = 0.6 is generated. What is the corresponding simulated value of X (to 2 decimal places)?
The simulated value is 18.33. Inverting F(x)=1-e^(-0.05x) gives x = -ln(1-u)/0.05, and with u = 0.6 this is -ln(0.4)/0.05 = 0.91629/0.05, which equals 18.33.
- A18.33Correct
- B10.22
- C13.86
- D9.16
- 32.19
Explanation
Set F(x)=u: x = -ln(1-u)/0.05. With u=0.6, ln(0.4) = -0.91629, so x = 0.91629/0.05 = 18.33. Using -ln(u)/0.05 gives 10.22, which is also valid in distribution but is a different mapping from the inverse transform of F at u, so it is not the value asked for. Options 13.86 and 9.16 use wrong rates or logs.
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