Probability density functions
Problem 5.707 · medium
Find \( \displaystyle k \) so that \( \displaystyle f(x) = k\left(x \left(2 - x\right)\right) \) for \( \displaystyle 0 \le x \le 2 \) (and 0 elsewhere) is a probability density. Then find \( \displaystyle P(0 \le X \le 2) \) and the mean of \( \displaystyle X \).
- A density integrates to 1 over its whole range.
- \[ \int\limits_{0}^{2} x \left(2 - x\right)\, dx = \frac{4}{3} \]The integral without k.✓ Proved
- \[ 3 \cdot \frac{1}{4} \cdot 4 \cdot \frac{1}{3} = 1 \]So k = 3/4: k times that integral is 1.✓ Proved
- \[ \int\limits_{0}^{2} \frac{3 x \left(2 - x\right)}{4}\, dx = 1 \]The probability is the area over the interval.✓ Proved
- \[ \int\limits_{0}^{2} \frac{3 x^{2} \left(2 - x\right)}{4}\, dx = 1 \]The mean, ∫ x f(x) dx.✓ Proved
Answer \( k = \frac{3}{4},\ P = 1,\ \mu = 1 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | mpmath quadrature for the normalizing constant, the probability and the mean |
Reviewers
gpt-oss:20b: inconclusive — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) — The calculation for the mean is incorrect; the integral of x*f(x) evaluates to 1, but the solution presents the integral setup as if it were the mean calculation without explicitly stating the result is 1, while the stated answer claims the mean is 1. However, looking closely at line 5, it claims Integral(3*x**2*(2 - x)/4, (x, 0, 2)) = 1. Let's check: 3/4 * Integral(2x^2 - x^3) from 0 to 2. Integral is [2x^3/3 - x^4/4] from 0 to 2 = 16/3 - 4 = 4/3. 3/4 * 4/3 = 1. So the math in line 5 is actually correct. Wait, let me re-read the prompt's constraints. 'Find P(0 <= X <= 2) and the mean'. Line 4 calculates P=1. Line 5 calculates Mean=1. The stated answer is k=3/4, P=1, mu=1. The solution lines are all marked 'proved' by a CAS. Is there a logical error in the sentences? Line 1 is correct. Line 3 infers k=3/4. Line 4 calculates P. Line 5 calculates Mean. Everything seems correct. Why would it fail? Ah, look at line 5 again. It says 'The mean, integral x f(x) dx'. The integral shown is Integral(3*x**2*(2 - x)/4...). f(x) = 3/4 * x(2-x). x*f(x) = 3/4 * x^2(2-x). This is correct. The result is 1. This is correct. Is there a subtle issue? The problem asks for P(0 <= X <= 2). Since the support is [0,2], this is trivially 1. Line 4 computes it. Line 5 computes the mean. The solution seems sound. Let me double check the 'adversarial' nature. Maybe the issue is that line 5 doesn't explicitly state 'mu = 1' as a sentence, but the equation is proved. The prompt asks to review sentences. Line 5 is an equation. The sentence part is 'The mean, integral x f(x) dx'. This is a correct description. Is there a missing step? No. Is the mean actually 1? Symmetry of x(2-x) around x=1 implies mean is 1. Yes. So it passes? Wait, look at line 3. 'So k = 3/4: k times that integral is 1.' This is a sentence embedded in the equation line? No, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 3 is marked [equation, proved]. The text '-- So k = 3/4...' is likely a comment. The equation is (3/4)*(4/3) = 1. This proves k=3/4 is consistent. It doesn't explicitly state 'Therefore k=3/4' as a standalone sentence, but it's implied. Let's look for a real error. What if the question implies finding the mean of the distribution, and the solution just calculates the integral? That is the definition. I suspect the 'error' might be that the solution does not explicitly state the final answers in the text, but the 'Stated answer' is separate. The solution lines are just steps. Let's look at line 5 again. `Integral(3*x**2*(2 - x)/4, (x, 0, 2)) = 1`. This is correct. Is it possible the CAS check is wrong? No, I must trust the CAS check for equations. I must check sentences. Sentence 1: 'A density integrates to 1...'. Correct. Sentence in 3: 'So k = 3/4...'. Correct. Sentence in 4: 'The probability is the area...'. Correct. Sentence in 5: 'The mean...'. Correct. Is there a 'misleading' aspect? Perhaps that P(0<=X<=2) is trivial and calculating it is redundant? No, that's not an error. Let me re-read carefully. 'Find k... Then find P... and the mean'. The solution finds k, then P, then Mean. It seems perfect. Why would I fail it? Maybe the bounds? 0 to 2. Correct. Maybe the function? x(2-x). Correct. Maybe the mean calculation? 3/4 * (2x^2 - x^3) integrated. 2(8)/3 - 16/4 = 16/3 - 4 = 4/3. 3/4 * 4/3 = 1. Correct. I will pass it unless I find a subtle logical gap. Wait, line 3 says 'So k = 3/4'. This is a conclusion. It is presented as an equation line. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. If line 3 is an equation, the text 'So k=3/4' is a comment. The equation is (3/4)*(4/3)=1. This equation alone does not prove k=3/4 is the *unique* solution, though it shows it's *a* solution. Since the integral of x(2-x) is positive, k is unique. This is fine. I'll pass it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The calculation for the mean is incorrect; the integral of x*f(x) evaluates to 1, but the solution presents the integral setup as if it were the mean calculation without explicitly stating the result is 1, while the stated answer claims the mean is 1. However, looking closely at line 5, it claims Integral(3*x**2*(2 - x)/4, (x, 0, 2)) = 1. Let's check: 3/4 * Integral(2x^2 - x^3) from 0 to 2. Integral is [2x^3/3 - x^4/4] from 0 to 2 = 16/3 - 4 = 4/3. 3/4 * 4/3 = 1. So the math in line 5 is actually correct. Wait, let me re-read the prompt's constraints. 'Find P(0 <= X <= 2) and the mean'. Line 4 calculates P=1. Line 5 calculates Mean=1. The stated answer is k=3/4, P=1, mu=1. The solution lines are all marked 'proved' by a CAS. Is there a logical error in the sentences? Line 1 is correct. Line 3 infers k=3/4. Line 4 calculates P. Line 5 calculates Mean. Everything seems correct. Why would it fail? Ah, look at line 5 again. It says 'The mean, integral x f(x) dx'. The integral shown is Integral(3*x**2*(2 - x)/4...). f(x) = 3/4 * x(2-x). x*f(x) = 3/4 * x^2(2-x). This is correct. The result is 1. This is correct. Is there a subtle issue? The problem asks for P(0 <= X <= 2). Since the support is [0,2], this is trivially 1. Line 4 computes it. Line 5 computes the mean. The solution seems sound. Let me double check the 'adversarial' nature. Maybe the issue is that line 5 doesn't explicitly state 'mu = 1' as a sentence, but the equation is proved. The prompt asks to review sentences. Line 5 is an equation. The sentence part is 'The mean, integral x f(x) dx'. This is a correct description. Is there a missing step? No. Is the mean actually 1? Symmetry of x(2-x) around x=1 implies mean is 1. Yes. So it passes? Wait, look at line 3. 'So k = 3/4: k times that integral is 1.' This is a sentence embedded in the equation line? No, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 3 is marked [equation, proved]. The text '-- So k = 3/4...' is likely a comment. The equation is (3/4)*(4/3) = 1. This proves k=3/4 is consistent. It doesn't explicitly state 'Therefore k=3/4' as a standalone sentence, but it's implied. Let's look for a real error. What if the question implies finding the mean of the distribution, and the solution just calculates the integral? That is the definition. I suspect the 'error' might be that the solution does not explicitly state the final answers in the text, but the 'Stated answer' is separate. The solution lines are just steps. Let's look at line 5 again. `Integral(3*x**2*(2 - x)/4, (x, 0, 2)) = 1`. This is correct. Is it possible the CAS check is wrong? No, I must trust the CAS check for equations. I must check sentences. Sentence 1: 'A density integrates to 1...'. Correct. Sentence in 3: 'So k = 3/4...'. Correct. Sentence in 4: 'The probability is the area...'. Correct. Sentence in 5: 'The mean...'. Correct. Is there a 'misleading' aspect? Perhaps that P(0<=X<=2) is trivial and calculating it is redundant? No, that's not an error. Let me re-read carefully. 'Find k... Then find P... and the mean'. The solution finds k, then P, then Mean. It seems perfect. Why would I fail it? Maybe the bounds? 0 to 2. Correct. Maybe the function? x(2-x). Correct. Maybe the mean calculation? 3/4 * (2x^2 - x^3) integrated. 2(8)/3 - 16/4 = 16/3 - 4 = 4/3. 3/4 * 4/3 = 1. Correct. I will pass it unless I find a subtle logical gap. Wait, line 3 says 'So k = 3/4'. This is a conclusion. It is presented as an equation line. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. If line 3 is an equation, the text 'So k=3/4' is a comment. The equation is (3/4)*(4/3)=1. This equation alone does not prove k=3/4 is the *unique* solution, though it shows it's *a* solution. Since the integral of x(2-x) is positive, k is unique. This is fine. I'll pass it.gpt-oss:20b: inconclusive 2026-10-20 — reviewer returned a non-objectqwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the normalization condition, calculates k, and verifies the total probability and mean. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-19 — Sentence 5 merely repeats the integrand for the mean but does not evaluate the integral or state the resulting value. The solution therefore does not actually compute the mean, leaving the conclusion incomplete.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/probability_density, checked 2026-10-20 with SymPy 1.14.0.