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pdf kx: library reserve time, find k

Short answer

Integrating kx from 0 to 2 gives 2k=1 so k=0.5. Then P(X≤1)=0.25, P(0.5≤X≤1.5)=0.50, and P(X>1.5)=0.4375.

Let X denotes the amount of time for which a book on two-hour reserve at a college library is checked out by a randomly selected students, and suppose that X has density function f(x) = kx, 0 ≤ x ≤ 2, 0 otherwise. a) Find the value of k b) Calculate P(X ≤ 1) c) Calculate P(0.5 ≤ X ≤ 1.5) d) Calculate P(1.5 < X)

a) Value of k

∫₀² kx dx = k [x²/2]₀² = 2k = 1, so k = 1/2. Hence f(x) = x/2 and the cdf is F(x) = x²/4 for 0 ≤ x ≤ 2.

b) P(X ≤ 1)

F(1) = 1/4 = 0.25

c) P(0.5 ≤ X ≤ 1.5)

F(1.5) − F(0.5) = 2.25/4 − 0.25/4 = 0.5625 − 0.0625 = 0.50

d) P(X > 1.5)

1 − F(1.5) = 1 − 0.5625 = 0.4375

Answer: k = 0.5; 0.25; 0.50; 0.4375.

✓ Answer checked by Claude Sonnet 5.5

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