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pdf Ax(6−x)²: find A, mean, variance

Short answer

Integrate f(x)=A x(6−x)^2 over 0–6, giving A=1/108. Mean μ = E(X) = 2.4. Variance = E(X^2)−μ^2 = 7.2−5.76 = 1.44.

A probability density function is given by f(x) = Ax(6 − x)² for 0 < x < 6 i) Find the value of A ii) Find the mean and variance of this distribution

i) Value of A

Expand x(6 − x)² = 36x − 12x² + x³.

∫₀⁶ (36x − 12x² + x³) dx = [18x² − 4x³ + x⁴/4]₀⁶ = 648 − 864 + 324 = 108

A × 108 = 1, so A = 1/108.

ii) Mean and variance

Mean: E(X) = (1/108) ∫₀⁶ x·(36x − 12x² + x³) dx = (1/108) ∫₀⁶ (36x² − 12x³ + x⁴) dx

= (1/108) [12x³ − 3x⁴ + x⁵/5] from 0 to 6
= (1/108) [2592 − 3888 + 1555.2]
= 259.2 / 108 = 2.4

E(X²) = (1/108) ∫₀⁶ (36x³ − 12x⁴ + x⁵) dx

= (1/108) [9x⁴ − 2.4x⁵ + x⁶/6] from 0 to 6
= (1/108) [11664 − 18662.4 + 7776]
= 777.6 / 108 = 7.2

Variance = E(X²) − [E(X)]² = 7.2 − (2.4)² = 7.2 − 5.76 = 1.44

Answer: A = 1/108, mean = 2.4, variance = 1.44 (standard deviation 1.2).

✓ Answer checked by Claude Sonnet 5.5

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