using-the-cdf-fx1-e-x3-for-x-0-derived-from-fx3x2-e-x3-compute-p1-x-3

Question

Using the CDF F(x)=1 - $$e^{-x^3}$$ for x ≥ 0 derived from f(x)=$$3x^2$$ $$e^{-x^3}$$, compute P(1 ≤ X ≤ 3).

Accepted Answer

Approximately 0.368 (exact form $$e^{-1}$$ - $$e^{-27})$$

Answer

Approximately 0.632 (which corresponds to 1 - $$e^{-1}$$, mistakenly ignoring the $$e^{-27}$$ term)

Answer

Approximately 0.000000000001879 because $$e^{-27}$$ dominates the numerator and the other exponential cancels out

Answer

Exactly 1 - $$e^{-3^3}$$ because probabilities over finite intervals with positive lower bound always simplify to one minus an exponential evaluated at the upper bound

Using the CDF F(x)=1 - e^{-x^3} for x ≥ 0 derived from f(x)=3x^2 e^{-x^3}, compute P(1 ≤ X ≤ 3).