Consider a probability distribution given by the density function P(x) = Cx² for 1 ≤ x ≤ 4, and 0 ot

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Question

Consider a probability distribution given by the density function P(x) = Cx² for 1 ≤ x ≤ 4, and 0 otherwise. The probability P(2 ≤ x ≤ 3) is ______. (rounded off to three decimal places)

Accepted Answer

Answer: 0.301Concept — Probability Density Function (PDF): For a continuous random variable, the total area under the PDF must equal 1 (normalization condition), and the probability of the variable falling in a range is the area under the PDF over that range.Step 1 — Find C using normalization:∫₁⁴ Cx² dx = 1C · [x³/3]₁⁴ = 1C · (64/3 − 1/3) = 1C · (63/3) = 1C · 21 = 1C = 1/21Step 2 — Compute P(2 ≤ x ≤ 3):P(2 ≤ x ≤ 3) = ∫₂³ (1/21)x² dx= (1/21) · [x³/3]₂³= (1/21) · (27/3 − 8/3)= (1/21) · (19/3)= 19/63≈ 0.30159 ≈ 0.301

Consider a probability distribution given by the density function P(x) = Cx² for 1 ≤ x ≤ 4, and 0 otherwise. The probability P(2 ≤ x ≤ 3) is ______. (rounded off to three decimal places)

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Consider a probability distribution given by the density function P(x) = Cx² for 1 ≤ x ≤ 4, and 0 otherwise. The probability P(2 ≤ x ≤ 3) is ______. (rounded off to three decimal places)

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