Exponential Distribution Calculator

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Parameters

Rate Parameter (λ)

Mean waiting time = 1/λ

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

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Exponential Distribution: Definition, Formula, and Applications

Exponential Distribution

Definition: The exponential distribution models the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Formula:The probability density function (PDF) and cumulative distribution function (CDF) are given by:

f(x) = \lambda e^{-\lambda x}, \quad x \geq 0

F(x) = 1 - e^{-\lambda x}, \quad x \geq 0

Where:

$\lambda$ is the rate parameter (average rate of events)
$x$ is the time between events

Properties

Mean (Expected Value): $E(X) = \frac{1}{\lambda}$
Variance: $\text{Var}(X) = \frac{1}{\lambda^2}$
Memoryless Property: $P(X > s + t | X > s) = P(X > t)$ . This means that if an event has not occurred for some time s, the probability of waiting an additional time t is the same as the probability of waiting time t from the start. This unique property means the distribution has no "memory" of past waiting time. For example, if a light bulb has been working for 10 hours, the probability it will work for one more hour is exactly the same as the probability a brand new bulb will work for one hour. Let's prove this mathematically: $P(X > s + t | X > s) = \frac{P(X > s + t)}{P(X > s)} = \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(X > t)$ This shows that waiting an additional time t after already waiting for time s has the same probability as waiting time t from the start.
Support: $[0, \infty)$
Right-skewed distribution
Decreasing failure rate

Applications

1. Reliability Engineering

The exponential distribution is widely used in reliability engineering to model the time until failure of electronic components, machinery, or entire systems. This is especially useful when the failure rate is constant over time, meaning that the likelihood of failure does not increase as the system ages. The memoryless property, unique to the exponential distribution, implies that past usage or operation time has no effect on future failure probability. This is useful for systems like light bulbs, electrical components, or mechanical parts where wear and tear doesn't impact failure in a predictable way.

2. Queueing Theory

In queueing theory, the exponential distribution is commonly used to model the time between events, such as the arrival of customers to a service point or the time between phone calls in a call center. The assumption that arrivals occur independently and randomly over time fits well with the exponential distribution. The memoryless property simplifies the analysis of these systems, as the future waiting time is unaffected by how long someone has already been waiting. This helps in analyzing average waiting times and optimizing service systems.

3. Physics

In physics, the exponential distribution plays a key role in modeling radioactive decay, where it describes the time between successive emissions of particles. The constant rate parameter in the distribution corresponds to the decay constant, which is characteristic of a particular radioactive substance. Since each decay event occurs independently of the time since the last event, the exponential distribution provides an accurate representation of the randomness in the decay process, making it a fundamental tool in nuclear physics.

4. Operations Research

Operations research uses the exponential distribution to model lead times in supply chain and inventory management. It helps businesses estimate the time between ordering a product and its delivery, allowing for more efficient inventory control. Additionally, in maintenance scheduling, the exponential distribution is used to model the time between breakdowns of equipment, enabling companies to develop optimal replacement and maintenance strategies. This ensures minimal downtime and cost-effective maintenance practices.

R Code Example

library(tidyverse)

# Calculate the probability of X between 1 and 3 (rate = 0.5)
P_between <- pexp(3, rate = 0.5) - pexp(1, rate = 0.5)
print(str_glue("P(1 < X < 3) = {round(P_between, 4)}"))

# X ~ Exp(0.25)
# Calculate the probability of X between 2 and 5
P_between <- pexp(5, rate = 0.25) - pexp(2, rate = 0.25)
print(str_glue("P(2 < X < 5) = {round(P_between, 4)}"))

# plot the exponential distribution with rate = 0.5
x <- seq(0, 8, length.out = 1000)
pdf <- dexp(x, rate = 0.5)
df <- tibble(x = x, pdf = pdf)

ggplot(df, aes(x = x, y = pdf)) +
  geom_line(color = "blue") +
  geom_area(data = subset(df, x >= 1 & x <= 3), aes(x = x, y = pdf), fill = "blue", alpha = 0.2) +
  labs(title = "Exponential Distribution (λ = 0.5)",
       x = "x",
       y = "Probability Density") +
  annotate("text", x = 4, y = 0.3, label = str_glue("P(1 < X < 3) = {round(P_between, 4)}"), hjust = 0) +
  theme_minimal()

Python Code Example

import numpy as np
import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt
import seaborn as sns

plt.style.use('seaborn')

# Calculate the probability of X between 1 and 3 (rate = 0.5)
P_between = stats.expon.cdf(3, scale=1/0.5) - stats.expon.cdf(1, scale=1/0.5)
print(f"P(1 < X < 3) = {P_between:.4f}")

# X ~ Exp(0.25)
# Calculate the probability of X between 2 and 5
P_between = stats.expon.cdf(5, scale=1/0.25) - stats.expon.cdf(2, scale=1/0.25)
print(f"P(2 < X < 5) = {P_between:.4f}")

# Plot the exponential distribution with rate = 0.5
x = np.linspace(0, 8, 1000)
pdf = stats.expon.pdf(x, scale=1/0.5)

# Create plot
plt.figure(figsize=(10, 6))
plt.plot(x, pdf, 'blue', label='PDF')

# Add shaded area for P(1 < X < 3)
x_shade = x[(x >= 1) & (x <= 3)]
pdf_shade = stats.expon.pdf(x_shade, scale=1/0.5)
plt.fill_between(x_shade, pdf_shade, alpha=0.2, color='blue')

# Add labels and title
plt.title('Exponential Distribution (λ = 0.5)')
plt.xlabel('x')
plt.ylabel('Probability Density')

# Add annotation for probability
P_between = stats.expon.cdf(3, scale=1/0.5) - stats.expon.cdf(1, scale=1/0.5)
plt.annotate(f'P(1 < X < 3) = {P_between:.4f}', 
            xy=(4, 0.3), 
            xytext=(4, 0.3),
            horizontalalignment='left')

plt.grid(True, alpha=0.3)
plt.show()