Support Region & Probability
Integration Setup
Closed-Form Result
◆ THE BLUEPRINT
What You're Looking At

The triangular region is the support of the joint PDF \(f(y_1, y_2) = 3y_1\) for \(0 \leq y_2 \leq y_1 \leq 1\). The shaded area represents \(P(Y_1 - Y_2 > c)\).

Key Definitions
\(f(y_1, y_2) \geq 0 \text{ for all } (y_1, y_2)\)
\(\int\!\!\int f(y_1, y_2)\,dy_1\,dy_2 = 1\)
Integration Order

Both integration orders yield the same probability (Fubini's theorem). The bounds change but the result does not.

dy₂ first: Fix y₁, integrate y₂ over its range. The outer integral sweeps y₁.

dy₁ first: Fix y₂, integrate y₁ over its range. The outer integral sweeps y₂.

Closed-Form
\(P(Y_1 - Y_2 > c) = 1 - \frac{3c}{2} + \frac{c^3}{2}\)
Related Topics
Marginals Tab 3 Conditional Tab 4
System Diagram
Joint PDF of Component Lifetimes
System Reliability R(t)
System Failure Density f(t)
MTTF
P(Survive to t=1)
Median Lifetime
◆ THE BLUEPRINT
What You're Looking At

Each component has an exponential lifetime with rate \(\lambda_i\). The system topology determines how component failures affect the system.

Exponential Reliability
\(R_i(t) = P(T_i > t) = e^{-\lambda_i t}\)
\(f_i(t) = \lambda_i e^{-\lambda_i t}, \quad t \geq 0\)
Series System

Fails if any component fails. "Weakest link."

\(R_{\text{series}}(t) = \prod_{i=1}^{n} R_i(t) = \prod_{i=1}^{n} e^{-\lambda_i t}\)
Parallel System

Fails only when all components fail. Redundancy.

\(R_{\text{parallel}}(t) = 1 - \prod_{i=1}^{n} [1 - R_i(t)]\)
Joint PDF

Component lifetimes are independent. The joint PDF is the product of individual exponential densities.

\(f(t_1, \ldots, t_n) = \prod_{i=1}^{n} \lambda_i e^{-\lambda_i t_i}\)
MTTF
\(\text{MTTF} = \int_0^{\infty} R_{\text{sys}}(t) \, dt\)
Related Topics
Marginals Tab 3 Conditional Tab 4
3D Joint Density Surface
Contour Plot with Marginals
Marginal Density
◆ THE BLUEPRINT
What You're Looking At

The 3D surface shows the joint density \(f(y_1, y_2)\). A marginal density is obtained by integrating out one variable. Visually, this "collapses" the surface onto one axis.

Marginal Density Functions
\(f_1(y_1) = \int_0^1 f(y_1, y_2) \, dy_2\)
\(f_2(y_2) = \int_0^1 f(y_1, y_2) \, dy_1\)
Interpretation

The marginal captures the unconditional behavior of a single variable. It ignores the value of the other variable entirely.

If the joint PDF factors as \(f(y_1, y_2) = f_1(y_1) \cdot f_2(y_2)\), the variables are independent. Otherwise they are dependent.

Related Topics
Triangle Region Tab 1 Conditional Tab 4
Conditionals Across Y₂
Independence Check
◆ THE BLUEPRINT
What You're Looking At

The joint PDF is \(f(y_1, y_2) = 1 + c(y_1 - \tfrac{1}{2})(y_2 - \tfrac{1}{2})\) on \([0,1]^2\). Both marginals are Uniform(0,1) for every value of \(c\).

The Fan Plot

The left plot shows the conditional density \(f(y_1 \mid y_2)\) evaluated at five different \(y_2\) values. When \(c = 0\), all five lines collapse to the same flat line. That is independence: knowing \(Y_2\) tells you nothing about \(Y_1\). When \(c \neq 0\), the lines fan out. Different \(y_2\) values produce different conditional shapes.

Conditional Density
\(f(y_1 \mid y_2) = \frac{f(y_1, y_2)}{f_2(y_2)}\)

Since \(f_2(y_2) = 1\) here, the conditional simplifies to \(f(y_1 \mid y_2) = f(y_1, y_2)\).

Independence
\(Y_1 \perp Y_2 \iff f(y_1, y_2) = f_1(y_1) \cdot f_2(y_2) \text{ for all } (y_1, y_2)\)

Equivalently, \(Y_1 \perp Y_2\) iff \(f(y_1 \mid y_2) = f_1(y_1)\) for all \(y_2\). The fan plot tests this visually.

3-Step Process

1. Find or identify the conditional density \(f(y_1 \mid y_2)\).

2. Plug in the conditioning value.

3. Integrate over the region of interest.

Related Topics
Marginals Tab 3 Sketch Lab Tab 5
◆ THE BLUEPRINT
How It Works

You are given a joint PDF and asked to sketch a marginal or conditional density. Click on the plot to place points that define your curve. Points connect left-to-right via linear interpolation.

Scoring

Your sketch is scored using the coefficient of determination (R\(^2\)). A score of 100 means a perfect match. A score of 0 means your sketch is no better than a flat line at the mean value.

\(\text{Score} = \max\left(0,\; 1 - \frac{\sum(\hat{f} - f)^2}{\sum(f - \bar{f})^2}\right) \times 100\)
Marginal Densities

To find \(f_1(y_1)\), integrate the joint PDF over all \(y_2\). Pay attention to the support region.

\(f_1(y_1) = \int f(y_1, y_2)\, dy_2\)
Conditional Densities

The conditional density requires dividing by the marginal.

\(f(y_1 | y_2) = \frac{f(y_1, y_2)}{f_2(y_2)}\)
Tips

Place points at the boundaries (x = 0 and x = 1) to anchor your curve. Think about whether the function should be increasing, decreasing, or curved before placing points.