Exponential Distribution

Exponential Distribution

A continuous probability distribution that models the time between independent events occurring at a constant average rate λ

Probability Density Function (PDF)

• f(x) = λe^-(λx) for x ≥ 0

• It describes how the probability is distributed over time intervals

• Used to model the time between successive arrivals at a service desk or failures of electronic components

Cumulative Distribution Function (CDF)

• F(x) = 1 - e^(-λx)

• Gives the probability that the waiting time is less than or equal to x

• In reliability engineering, F(x) represents the probability that a component will fail before time x.

Expected Value (Mean)

• μ_Χ = 1/λ

• The average waiting time between events

• If the average rate of incoming network packets is 5 per second, the mean waiting time between arrivals is 0.2 seconds

Variance

• (σ_Χ)² = (1/λ²)

• Measures the spread of waiting times

• A higher λ (faster process) results in smaller variance, indicating more predictable waiting times

Standard Deviation

• σ_X = 1/λ

• Equals to the mean

• Showing proportional speed

• If average bulb lifetime is 1000 hours, its standard deviation is also 1000 hours.

Memoryless Property

• P(X > s + t|X > s) = P(X > t)

• The future waiting time does not depend on how much time has already elapsed

• For radioactive decay, the chance an atom survives another 10 decay is independent of how long it has already existed

Exponential distribution’s relationship to the Poisson Process

• The exponential distribution models the waiting time between events in a Poisson process with rate parameter λ

• If call arrive at a call center following a Poisson process with λ = 3 per minute

• The waiting time between calls follows Exp(3)

Derivation of the Mean

• μ_X = ∫(from 0 to ∞) xλe^-(λx) dx = 1/λ

• Evaluated by the integration by parts and L’Hôpital’s rule

• to show the term -xe^-(λx) approaches zero as x → ∞

Derivation of the Variance

• (σ_Χ)² = ∫(from 0 to ∞) x²λe^-(λx) dx - (μ_X)² = 1/(λ²)

• Integration by parts confirms the result

• with boundary terms vanishing as x → ∞

The Relationship Between Exponential and Poisson Distributions

• If T is the waiting time until the next event in a Poisson process with rate λ

• Then T ~ Exp(λ)

• P(T > t) = P(X = 0) = e^-(λt) gives us:

• F(t) = 1 - e^-(λt)

Probability of Waiting Longer than t

• P(T > t) = e^-(λt)

• If server crashes occur at λ = 0.2 per hour,

• The probability of running more than 10 hours without a crash will be:

• e^-2 = 0.135

Propagation of Uncertainty in λ ̂ (Estimated Rate)

• σ_λ ̂ = | d/dX̄ 1/X̄ | σ_X̄ = 1/( X̄ n^1/2)

• Used to quantify how precisely λ (event rate) is estimated from n observed intervals

Standard Error of the Mean for Exponential Distribution

• σ_X̄ = σ/(n^1/2) = 1/(λ*n^1/2)

• If mean lifetime of 10 devices is measured, this gives uncertainty in the average.

Estimation of Rate Parameter λ

• λ ̂ = 1/X̄

• The rate parameter is the reciprocal of the sample mean

• In system reliability,

• If average time between failure is 50 hours,

• Estimated failure rate is 0.02 failures/hour

Uncertainty in Estimated Rate (σ_λ ̂)

• σ_λ ̂ = 1/(X̄*(n^1/2))

• For 25 observed events with average waiting time 2 minutes

• σ_λ ̂ = 1/(2 × 5) = 0.1 min^-1

Component Lifetimes (Engineering Example)

• The exponential distribution models how long a component operates before failure when the failure rate is constant.

• For an electronic sensor with mean life 5000 hours

• P(failure before 1000h) = 1 - e^(-1000/5000) = 0.181

Quality Testing (Industrial Example)

• Used to predict the waiting time between defective products on an assembly line.

• If defects occur on average every 200 items, λ = 1/200, so the chance no defect occurs in 300 items is e^-1.5 = 0.22

Communication Systems Example

• Models the time between successive data packets or bit errors in a communication link.

• If the average arrival packer rate is 1000 packets/sec

• Then the waiting time between packets follows Exp(1000)

Time to Next Event (Medical Example)

• Represents waiting time until the next biological event such as infection, decay, or heart beat irregularity.

• If the rate of heartbeat irregularities is 0.05 per minute, the waiting time follows Exp(0.05)