Consider the probability distribution function defined as f_X(x) = |x|, & |x| < 1 \\ 0, & otherwise . Which of the following is true? - Mathematics MCQ Mathematics Mock Test - 9 2026 | ExamTest.live

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Consider the probability distribution function defined as f_{X} (x) = {∣ x ∣, 0, ∣ x ∣ < 1 otherwise . Which of the following is true?

Solution & Step-by-step Explanation

For a function to be a valid PDF: f_{X} (x) \geq 0 for all x .Here ∣ x ∣ \geq 0, so this condition is met. \int_{- \infty}^{\infty} f_{X} (x) d x = 1 . \int_{- 1}^{1} ∣ x ∣ d x = \int_{- 1}^{0} - x d x + \int_{0}^{1} x d x = [- \frac{x ^{2}}{2}]_{- 1}^{0} + [\frac{x ^{2}}{2}]_{0}^{1} = (0 - (- 1/2)) + (1/2 - 0) = 1/2 + 1/2 = 1 .Both conditions are satisfied, so it is a valid PDF.

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Consider the probability distribution function defined as f_{X} (x) = {∣ x ∣, 0, ∣ x ∣ < 1 otherwise . Which of the following is true?

Invalid probability distribution function

Valid probability distribution function

f_{X} (x) < 0

Can't say

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