TABLE OF CONTENTS

Probability II

VOL. IMAR 2026

CH1 — Part 2

PDF Derivations and Area Accumulation

Example 3: Normalisation & Derivation

If $X$ is a random variable having the pdf of the form:

f_X(x) = \begin{cases} cx & 0 < x < 2 \\ 0 & \text{otherwise} \end{cases}

Example 3 (a)

Find c

Find the constant $c$ that makes $f_X(x)$ a valid pdf.

PDF Normalization

Total Area0.60

Control Constant

c

c = 0.30

3 (b) Deriving the Piecewise CDF

To find $F_X(x)$ , we integrate the piecewise density function $f_X(t)$ from $-\infty$ to $x$ across all three mathematical regions.

Derivation Trace

Use the slider to see how the integral accumulates over each mathematical region.

Case 1: -1 ≤ x < 0

\int_{-\infty}^x 0\,dt

0

Density f(t)

Cumulative F(x)

Evaluation

F(-1.00) = 0.000

x = -1.0evaluation x = -1.00x = 3.0

Complete Definition Result

F_X(x) = \begin{cases} \color{#9b2c1c} 0 & x < 0 \\ \tfrac{1}{4}x^2 & 0 \leq x < 2 \\ 1 & x \geq 2 \end{cases}

Example 4: Area Calculation

Now that we have the PDF and CDF for Example 3, we can calculate the probability that $X$ falls between 1 and 2. This interactive visualization shows both the Integration Method (area under the PDF) and the Subtraction Method (CDF evaluation).

Example 4

P(1 < X < 2)

Find $P(1 < X < 2)$ .

Probability as Area

a = -1.00

b = 1.00

Bound A

Bound B

Integration Method

P(-1.0 \leq X \leq 1.0) = \int_{-1.0}^{1.0} f_X(x)\,dx \approx 0.6827

CDF Subtraction Method

F_X(1.0) - F_X(-1.0) = 0.8413 - 0.1587 = 0.6827

Probability as Area

Remark — Area Interpretation

For a continuous random variable, the probability $P(a \leq X \leq b)$ is exactly the area under the density curve between $a$ and $b$ .

P(a \leq X \leq b) = \int_a^b f_X(x)\,dx = F_X(b) - F_X(a)

Interval Equivalence

Because $P(X=a)=0$ and $P(X=b)=0$ for continuous variables, all the following probabilities are equal:

P(a < X < b)

P(a \leq X < b)

P(a < X \leq b)

P(a \leq X \leq b)

Example 5: The Tent Function

This example demonstrates the Fundamental Theorem of Calculus in its piecewise form. Notice how the continuity of $F_X(x)$ is preserved at the boundaries.

The "tent" function is a triangular density symmetric around $x=1$ :

f_X(x) = \begin{cases} x & 0 < x < 1 \\ 2-x & 1 < x < 2 \\ 0 & \text{otherwise} \end{cases}

Example 5

Comprehensive Analysis

Find (a) $F_X(x)$ , (b) $P(X \leq 0.2)$ , (c) $P(X \leq 1.5)$ , (d) $P(0.2 < X < 1.5)$ , (e) $P(0.1 < X \leq 0.9)$ , (f) $P(X > 6)$ .

Interactive Visualisation

Building the Tent CDF

0 ≤ x < 1

Density

f(x) = x

Cumulative

F(x) = x^2/2

1 ≤ x < 2

Density

f(x) = 2 - x

Cumulative

F(x) = 1 - \frac{(2-x)^2}{2}

Direct Manipulation Active

Current Value

x = -0.20

Density Function Trace

Cumulative Distribution Trace

x = -0.20

Calculus Computation

Integration Result

F_X(-0.20) = \int_{-\infty}^{-0.20} f_X(t)\,dt = 0.0000

Region 0Live evaluation

Exercises

Exercise 1

Let $f_X(x) = 2$ for $1 \leq x \leq 1.5$ , and $0$ otherwise. Is this a valid pdf?

Exercise 2

If $f_X(x) = cx^3(1-x)$ for $0 \leq x \leq 1$ .
(a) Find c.
(b) Find $P(X \leq 1/2)$ .
(c) Find $P(X \geq 1/2)$ .