Continuous Cumulative Distribution Function (CDF) | Probability & Statistics

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Definition: Continuous CDF

A continuous random variable can take an infinite number of values in a given range.
The Probability Density Function (PDF) \( f(x) \) describes the likelihood of \( X \) falling within a small interval.
The Cumulative Distribution Function (CDF) is given by:
\[
F(x) = P[X \leq x] = \int_{-\infty}^{x} f(t) dt
\]
The function \( F(x) \) is non-decreasing and satisfies:
\[
\lim_{x \to -\infty} F(x) = 0, \quad \lim_{x \to \infty} F(x) = 1
\]

The diameter \(X\) of a cable is a continuous random variable with probability density function (PDF):

\[
f(x) = \begin{cases}
6x(1-x), & 0 \leq x \leq 1 \\
0, & \text{otherwise}
\end{cases}
\]

Find the cumulative distribution function \( F(x) \).

The cumulative distribution function (CDF) is given by:

\[
F(x) = P[X \leq x] = \int_{-\infty}^{x} f(t) dt
\]

Since \( f(x) = 0 \) for \( x < 0 \), we integrate:

\[
F(x) = \int_0^x 6t(1 – t) dt
\]

Computing:

\[
\int_0^x (6t – 6t^2) dt
\]

Evaluating the integral:

\[
\left[ 3t^2 – 2t^3 \right]_0^x = 3x^2 – 2x^3
\]

Final result:

\[
F(x) = \begin{cases}
0, & x < 0 \\ 3x^2 – 2x^3, & 0 \leq x \leq 1 \\ 1, & x > 1
\end{cases}
\]