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- 2073 Shrawan · 5 marks
pdf x/2: library reserve time
Short answer
The cdf is F(x)=x²/4 for 0≤x≤2. P(X≤1)=F(1)=0.25, and P(0.5≤X≤1.5)=F(1.5)−F(0.5)=0.50.
Let X denote the amount of time for which a book on two hour reserve at a college library is checked out by a randomly selected student and suppose that X has density function f(x) = (1/2)x for 0 ≤ x ≤ 2, 0 otherwise. Calculate P(X ≤ 1) and P(0.5 ≤ X ≤ 1.5).
Check: ∫₀² (x/2) dx = [x²/4]₀² = 1, so f(x) is a valid pdf. The cdf for 0 ≤ x ≤ 2 is F(x) = ∫₀ˣ (t/2) dt = x²/4.
P(X ≤ 1)
P(X ≤ 1) = F(1) = 1²/4 = 0.25
P(0.5 ≤ X ≤ 1.5)
P = F(1.5) − F(0.5) = (1.5)²/4 − (0.5)²/4 = 2.25/4 − 0.25/4 = 0.5625 − 0.0625 = 0.50
Answer: P(X ≤ 1) = 0.25 and P(0.5 ≤ X ≤ 1.5) = 0.50.
✓ Answer checked by Claude Sonnet 5.5
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