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UPCET / UPSEE Cumulative Distribution Function — practice questions

Explore 3 free Cumulative Distribution Function questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Cumulative Distribution Function questions and exercises with answers

Showing questions 1–3 of 3.

The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:Read the answer and worked solution →The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:Read the answer and worked solution →For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:Read the answer and worked solution →

Worked practice questions

The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:

  1. $F(x) = P[X = x]$
  2. $F(x) = 1 - p$
  3. $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$
  4. $F(x) = P[X > x]$
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Correct answer: C. $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$

c.d.f. accumulates: $F(x) = P[X \leq x] = \sum_{x_i \leq x} p_i$. It is non-decreasing, starts at 0 (for $x < \min$), reaches 1 (at $x \geq \max$). For discrete RV it's a step function.

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The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:

  1. $3x$
  2. $6x$
  3. $x^3$
  4. $1/(3x^2)$
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Correct answer: C. $x^3$

$F(x) = \int_0^x 3t^2\,dt = [t^3]_0^x = x^3$ for $0 \leq x \leq 1$.

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For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:

  1. $1$
  2. $0$
  3. $2.0$
  4. $1.25$
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Correct answer: D. $1.25$

$F(0.5) = 3(0.5) - 2(0.5)^3 = 1.5 - 2(0.125) = 1.5 - 0.25 = 1.25$. (Note: this exceeds 1, which signals a problem with the source p.d.f. — but the question is purely about evaluating the given $F$ formula.)

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