VOL. IMAR 2026

Institutional

VOL. IMAR 2026

The Imprint Engine.

We partner with university mathematics and statistics departments to transform static PDF syllabi into living, interactive digital infrastructure.

"The medium is the message. A static medium teaches a static science."

— Marshall McLuhan

Traditional statistical education relies on a broken medium. Students are handed static PDFs containing complex derivations, and expected to build dynamic, multidimensional intuition from flat text.

Show, Don't Tell

We take your existing lecture notes, problem sets, and syllabi, and re-engineer them into browser-native interactive experiences.

Static Past vs. Live Future

Scroll down to watch static PDF derivations wake up and physically transform into live, interactive learning components.

Chapter 1: Probability Density

If $F_X(x)$ is the cdf of a continuous r.v. $X$ , its derivative is called the probability density function (pdf):

f_X(x) = \frac{d}{dx} F_X(x)

Which implies the Fundamental Theorem relationship:

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

The PDF is not a probability. $f_X(x)$ can exceed 1. What must equal 1 is the total area under the curve.

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

Interactive Engine

Continuous Random Variables

Probability Density

f_X(t)

Cumulative Distribution

F_X(x)

Adjust Value

x

x = 0.00

-303

Integration

P(X \le 0.00) = \int_{-\infty}^{0.00} f_X(t) \, dt

Cumulative Value & Confidence

F_X(0.00) = 0.5000

Confidence: 50.0%

Chapter 2: Deriving the CDF

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

F_X(x) = \begin{cases} \int_{-\infty}^x 0\,dt & x < 0 \\ F_X(0) + \int_0^x \frac{1}{2}t\,dt & 0 \leq x < 2 \\ F_X(2) + \int_2^x 0\,dt & x \geq 2 \end{cases}

F_X(x) = \begin{cases} \int_{-\infty}^x 0\,dt & x < 0 \\ F_X(0) + \int_0^x \frac{1}{2}t\,dt & 0 \leq x < 2 \\ F_X(2) + \int_2^x 0\,dt & x \geq 2 \end{cases}

Interactive Engine

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}

Live Mathematics

LaTeX equations rendered natively alongside interactive parameter sliders. Students manipulate the math and see the geometry change instantly, closing the gap between symbolic logic and visual intuition.

Parameter Control

Mean (μ)0.0

Standard Deviation (σ)1.0

Bilingual Architecture

Full support for English and Arabic (RTL) technical typesetting. Deploy the same rigorous course to international and domestic cohorts seamlessly, maintaining perfect mathematical alignment.

Typesetting Engine

The variance of a continuous random variable $X$ with density $f(x)$ is given by:

\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx

Commissioned Prototypes

Live Demo Server

Cairo University (FEPS)

Spring 2021

Probability II

A rigorous treatment of continuous probability measures, derived from the Spring 2021 lectures by Prof. Reda Mazloum.

Cairo University (FEPS)

Spring 2021

Theory of Statistics 1

A rigorous exploration of discrete variable transformations, derived from the Spring 2021 lectures by Prof. Reda Mazloum.

Commission an Imprint.

Inquire about translating your department's curriculum into our interactive engine.

Contact Partnerships