X has density c₁/√x for 0<x≤1,c₂/x² for x>1,zero otherwise. If P(1/4≤X≤4)=5/8, consider I:P(3≤X≤5)=1/12. II:neither X nor 1/X has finite expectation. Which hold?
Normalization gives 2c₁+c₂=1, and the supplied interval probability gives c₁+(3/4)c₂=5/8. Thus c₁=1/4,c₂=1/2. The probability in I is (1/2)(1/3−1/5)=1/15, not 1/12. E X diverges at infinity like ∫dx/x, and E(1/X) diverges at zero like ∫x^(−3/2)dx. Only II, **B**.
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