FRM Part I · FRM Exam Part I · Random Variables
A continuous random variable X has probability density function f(x) = 2x for 0 ≤ x ≤ 1 and 0 elsewhere. What is P(0.5 < X ≤ 0.8)?
For a continuous variable, probability is the area under the density. Integrating 2x gives the CDF x squared, so P(0.5 < X ≤ 0.8) = 0.64 - 0.25 = 0.39. Treating the density as a probability or ignoring the lower bound gives wrong answers.
- A0.30
- B0.39Correct
- C0.60
- D0.64
Explanation
The probability is the integral of f from 0.5 to 0.8, which equals x^2 evaluated between those limits: 0.64 - 0.25 = 0.39. The value 0.30 wrongly multiplies the interval width by a density of 1. The value 0.64 is P(X ≤ 0.8), which omits subtracting P(X ≤ 0.5).
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