FRM Part I · FRM Exam Part I · Fundamentals of Probability
A continuous random variable X has probability density function f(x) = 2x for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. What is P(0.5 < X ≤ 0.8)?
Integrating the density 2x gives the cumulative distribution function F(x) = x squared. The probability of falling between 0.5 and 0.8 is F(0.8) minus F(0.5) = 0.64 - 0.25 = 0.39.
- A0.39Correct
- B0.30
- C0.64
- D0.25
Explanation
The CDF is F(x) = x^2 on [0,1]. P(0.5 < X ≤ 0.8) = 0.8^2 - 0.5^2 = 0.64 - 0.25 = 0.39. Using 0.64 gives only F(0.8), ignoring the lower bound; 0.30 wrongly multiplies the interval width 0.3 by 1 as if density were uniform.
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