📘 4.2 – Cumulative Distribution Function (CDF)

🔍 What is a CDF?

The cumulative distribution function (CDF) tells us the probability that a random variable \( X \) is less than or equal to a value \( x \):

\[ F(x) = P(X \le x) = \int_{-\infty}^{x} f(u)\,du \]

The CDF starts at 0 and increases to 1.
It is non-decreasing for continuous variables.
The derivative of the CDF is the PDF: \( f(x) = \dfrac{d}{dx}F(x) \).

Example: Uniform Distribution (Interactive)

Let \(X \sim \text{Uniform}(a,b)\) with PDF

\[ f(x)=\begin{cases} \dfrac{1}{b-a}, & a\le x\le b\\[6pt] 0, & \text{otherwise} \end{cases} \]

Then the CDF is

a b k

Live result:

PDF \(f(x)\) with area \(P(X\le k)\) shaded

CDF \(F_X(k)\)

✅ Example Calculation

Question: What is \(F_X(k)=P(X\le k)\)?

Show Answer (with steps)

🧠 Practice Question – GRE Style (Interactive)

The time to finish a math section (minutes) is modeled by an exponential distribution with rate \(\lambda=0.04\ \text{per minute}\).
PDF: \(f(t)=\lambda e^{-\lambda t}\) for \(t\ge0\). CDF: \(F(t)=P(T\le t)=1-e^{-\lambda t}\) for \(t\ge0\).

λ (rate, min⁻¹) t (minutes) Range max (x-axis)

Live result:

PDF \(f(t)=\lambda e^{-\lambda t}\). Area up to \(t\) (shaded) = \(F(t)\).

CDF \(F(t)=1-e^{-\lambda t}\) with point at your \(t\).

📖 Show Answer (with very detailed steps)

Click to expand

✅ Summary