Exponential Distribution Probability Calculator

Exponential Distribution Calculator

Use this calculator to find probabilities or critical values for the Exponential distribution.

Rate Parameter (λ):

Select Calculation Type:

Score (X):

Probability (P(X ≤ x)):

Score (X₂):

Probability (P(X ≤ x₂)):

P(X ≤ x₁):

P(X > x₁):

Exponential Density (f(x₁)):

Score (x₁):

Score (x₂):

P(X ≤ x₂):

P(X > x₂):

Exponential Density (f(x₂)):

P(x₁ ≤ X ≤ x₂):

Understanding the Exponential Distribution

The Exponential distribution is a continuous probability distribution commonly used to model the time between independent events that occur at a constant average rate. It is often applied in fields like reliability analysis, queuing theory, and in modeling the lifetimes of objects or waiting times for certain events. The Exponential distribution is right-skewed and defined only for non-negative values (0 to ∞).

Key Components of the Exponential Distribution

Rate Parameter (λ):
- λ (lambda): The rate at which events occur, which is the reciprocal of the mean time between events. For example, if an event occurs on average every 2 minutes, then \( \lambda = 0.5 \) events per minute.
Probability (P): The probability that a random variable following the Exponential distribution is less than or equal to a certain value (X).
Score (X): A specific value within the Exponential distribution used to calculate probabilities, representing, for example, the time until an event occurs.

Exponential Distribution Formula

The probability density function (pdf) of the Exponential distribution with rate parameter \( \lambda \) is given by:

\[ f(x; \lambda) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 \]

This formula expresses the probability density for each possible time \( x \) until the next event. The cumulative distribution function (CDF) for the Exponential distribution, which gives the probability that \( X \) is less than or equal to \( x \), is:

\[ P(X \leq x) = 1 - e^{-\lambda x} \quad \text{for } x \geq 0 \]

Conditions for Using the Exponential Distribution

Memorylessness: The Exponential distribution is memoryless, meaning the probability of an event occurring in the next interval is independent of how much time has already passed. This property is unique to the Exponential distribution and is mathematically represented as \( P(X > s + t \,|\, X > s) = P(X > t) \).
Constant Rate: Events must occur at a constant average rate, implying that the time between events does not change over time.
Independence: The time between consecutive events must be independent of other time intervals.

Calculating Exponential Distribution Probabilities and Values

To compute Exponential distribution probabilities and values, we can use libraries in JavaScript, Python, and R. Here’s how to perform these calculations:

1. Using JavaScript

In JavaScript, you can use the jStat library for probability and inverse probability calculations.

// Calculate probability (CDF) for a given X
const lambda = 1.5; // Rate parameter
const x = 2.5; // Time until the event

// CDF: P(X ≤ x)
const cumulativeProbability = jStat.exponential.cdf(x, lambda);

// Inverse CDF: Get the time for a given probability (e.g., 0.95)
const probability = 0.95;
const timeForProbability = jStat.exponential.inv(probability, lambda);

console.log('Cumulative Probability:', cumulativeProbability);
console.log('Time for P(X ≤ x) = 0.95:', timeForProbability);

Note: Ensure you have the jStat library included in your project to perform these calculations.

2. Using Python

In Python, the SciPy library provides functions for cumulative probability and inverse calculations.

from scipy.stats import expon

# Define the rate parameter
lambda_ = 1.5  # Rate parameter
x = 2.5  # Time until the event

# CDF: P(X ≤ x)
cumulative_probability = expon.cdf(x, scale=1/lambda_)

# Inverse CDF: Get the time for a given probability (e.g., 0.95)
probability = 0.95
time_for_probability = expon.ppf(probability, scale=1/lambda_)

print("Cumulative Probability:", cumulative_probability)
print("Time for P(X ≤ x) = 0.95:", time_for_probability)

Note: Install SciPy by running pip install scipy if it’s not already installed.

3. Using R

In R, the stats package includes functions to calculate Exponential distribution probabilities and values.

# Define the rate parameter
lambda <- 1.5  # Rate parameter
x <- 2.5  # Time until the event

# CDF: P(X ≤ x)
cumulative_probability <- pexp(x, rate=lambda)

# Inverse CDF: Get the time for a given probability (e.g., 0.95)
probability <- 0.95
time_for_probability <- qexp(probability, rate=lambda)

cat("Cumulative Probability:", cumulative_probability, "\n")
cat("Time for P(X ≤ x) = 0.95:", time_for_probability, "\n")

Note: The pexp function calculates the cumulative probability, while qexp gives the time for a specified probability.