Subtopic: Mean, median, mode and standard deviation
Question Statement:
In a locality 'A', the probability of a convective storm event is 0.7 with a probability density function $f_{X_1}(x) = 0.1 e^{-0.1x}$, and the probability of a frontal storm event is 0.3 with a density function $f_{X_2}(x) = 0.2 e^{-0.2x}$ ($x > 0$). The combined probability that storm depth $X$ exceeds 10 mm is ______
Official Answer Key:
0.28 to 0.32
Step-by-Step Solution & Explanation
The storm magnitude X is a mixture: X=X₁ with probability 0.7 (exponential, rate 1) or X=X₂ with probability 0.3 (exponential, rate 2; note P(convective)+P(cyclone)=1 ⇒ P(cyclone)=0.3). For an exponential distribution with rate λ, P(X>1)=e^(−λ). So:
P(X>1) = 0.7×P(X₁>1) + 0.3×P(X₂>1) = 0.7×e⁻¹ + 0.3×e⁻²
= 0.7×0.36788 + 0.3×0.13534 = 0.25752 + 0.04060 = 0.2981 ≈ 0.30. Within official range 0.28–0.32.
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