Explore Exponential Distribution with Ease

Exponential Distribution Probability Density Function

Understanding Exponential Distribution

The Exponential Distribution models the time until an event occurs in a Poisson process (events happening continuously and independently at a constant average rate). It's memoryless, meaning past time doesn't affect future probabilities. It's defined by a rate parameter (\( \lambda \)). Common uses include modeling device lifetimes or waiting times.

Key Formulas:

• Probability Density Function (PDF): \( f(t; \lambda) = \lambda e^{-\lambda t}, \quad t \geq 0 \)
• Cumulative Distribution Function (CDF) for P(X \( \leq t \)): \( P(X \leq t) = 1 - e^{-\lambda t}, \quad t \geq 0 \)
• Probability for P(X > \( t \)): \( P(X > t) = e^{-\lambda t}, \quad t \geq 0 \)

  Use this tool to calculate probabilities and visualize the distribution for better understanding. .interactive-container { @apply shadow-xl rounded-2xl bg-background p-8 space-y-8; } .input-field { @apply space-y-2; } .input-label { @apply block text-sm font-medium text-default mb-1; } .input-box { @apply shadow-sm bg-input border border-gray-300 text-default text-sm rounded-lg focus:ring-primary focus:border-primary block w-full p-3; } .input-tip { @apply mt-1 text-gray-500 text-xs italic; } .selection-field { @apply space-y-2 mt-4; } .select-box { @apply bg-input border border-gray-300 text-default text-sm rounded-lg focus:ring-primary focus:border-primary block w-full p-3; } .action-buttons { @apply mt-6 flex justify-start space-x-4; } .primary-button { @apply text-foreground bg-background hover:bg-background focus:ring-4 focus:outline-none focus:ring-primary font-medium rounded-lg text-sm px-5 py-3 text-center disabled:bg-background disabled:cursor-wait; } .secondary-button { @apply text-gray-900 bg-gray-300 hover:bg-gray-400 focus:ring-4 focus:outline-none focus:ring-gray-200 font-medium rounded-lg text-sm px-5 py-3 text-center; } .output-box { @apply mt-8 p-6 border rounded-md bg-input shadow-sm; } .output-header { @apply flex items-center justify-between mb-4; } .output-title { @apply text-lg font-semibold text-default; } .output-value { @apply text-2xl font-bold text-foreground mt-2; } .copy-button { @apply bg-gray-200 hover:bg-gray-300 text-gray-800 font-medium rounded-md text-sm px-3 py-1; } .visualization-box { @apply mt-8; } .visualization-title { @apply text-xl font-semibold text-center text-default mb-4; } .plotly-chart { @apply w-full h-96; /* Adjust height as needed */ } .docs-section { @apply max-w-5xl mx-auto mt-16 p-8 bg-background rounded-2xl shadow-lg; } .docs-title { @apply text-2xl font-bold text-default mb-4; } .docs-text { @apply text-gray-700 leading-relaxed; }