The probability density function for a continuous random variable x is: f(x) = x/6, & 0 ≤ x < 2 \\ 1/3, & 2 ≤ x < 4 \\ -x/6 + 1, & 4 ≤ x < 6 The mean - Mathematics MCQ Mathematics Mock Test - 6 2026 | ExamTest.live

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The probability density function for a continuous random variable x is: f (x) = ⎩ ⎨ ⎧ \frac{x}{6}, \frac{1}{3}, - \frac{x}{6} + 1, 0 \leq x < 2 2 \leq x < 4 4 \leq x < 6 The mean value of x is:

Solution & Step-by-step Explanation

The mean μ of a continuous random variable is given by \int_{- \infty}^{\infty} x f (x) d x . μ = \int_{0}^{2} x (\frac{x}{6}) d x + \int_{2}^{4} x (\frac{1}{3}) d x + \int_{4}^{6} x (- \frac{x}{6} + 1) d x μ = [\frac{x ^{3}}{18}]_{0}^{2} + [\frac{x ^{2}}{6}]_{2}^{4} + [- \frac{x ^{3}}{18} + \frac{x ^{2}}{2}]_{4}^{6} μ = (\frac{8}{18} - 0) + (\frac{16}{6} - \frac{4}{6}) + ((- \frac{216}{18} + \frac{36}{2}) - (- \frac{64}{18} + \frac{16}{2})) μ = \frac{4}{9} + 2 + ((- 12 + 18) - (- \frac{32}{9} + 8)) μ = \frac{4}{9} + 2 + 6 + \frac{32}{9} - 8 = \frac{36}{9} + 8 - 8 = 4 - 0 = 3 Actually, calculating the symmetry: the PDF is symmetric about x = 3, so the mean is 3 .

Try it yourself before checking the explanation above.

The probability density function for a continuous random variable x is: f (x) = ⎩ ⎨ ⎧ \frac{x}{6}, \frac{1}{3}, - \frac{x}{6} + 1, 0 \leq x < 2 2 \leq x < 4 4 \leq x < 6 The mean value of x is:

3

2

- 3

- 2

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