pdf definition; f(x)=Kx(1−x) probabilities

Probability density function

For a continuous random variable X, the pdf f(x) is a function such that

1. f(x) ≥ 0 for all x,
2. ∫ f(x) dx = 1 over the whole range, and
3. P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.

Find K

f(x) = Kx(1 − x), 0 ≤ x ≤ 1.

∫₀¹ K(x − x²) dx = K(1/2 − 1/3) = K/6 = 1 → K = 6

So f(x) = 6x(1 − x), and the cumulative function is F(x) = 6(x²/2 − x³/3) = 3x² − 2x³.

(i) P(X ≥ 0.4)

P = 1 − F(0.4) = 1 − [3(0.16) − 2(0.064)] = 1 − (0.48 − 0.128) = 1 − 0.352 = 0.648

(ii) P(1/4 ≤ X ≤ 3/4)

F(0.75) = 3(0.5625) − 2(0.421875) = 1.6875 − 0.84375 = 0.84375 F(0.25) = 3(0.0625) − 2(0.015625) = 0.1875 − 0.03125 = 0.15625

P = 0.84375 − 0.15625 = 0.6875

Answer: K = 6, (i) 0.648, (ii) 0.6875