Exponential and gamma distributions are key players in modeling waiting times and event occurrences. The exponential distribution focuses on the time between events, while the gamma distribution extends this to multiple events, making them crucial for reliability analysis and queueing theory.

These distributions showcase the versatility of continuous probability models. With their unique properties like memorylessness for exponential and shape flexibility for gamma, they provide powerful tools for analyzing real-world phenomena in fields ranging from engineering to finance.

Exponential and Gamma Distributions

Exponential Distribution Fundamentals

• Exponential distribution models time between events in a Poisson process
• Probability density function (PDF) of exponential distribution $f(x) = λe^{-λx}$ for x ≥ 0, where λ represents the rate parameter
• Cumulative distribution function (CDF) $F(x) = 1 - e^{-λx}$ for x ≥ 0
• Expected value (mean) of exponential distribution $E(X) = \frac{1}{λ}$
• Variance of exponential distribution $Var(X) = \frac{1}{λ^2}$
• Memoryless property unique to exponential distribution means future waiting time independent of elapsed time

Gamma Distribution and Its Relationship to Exponential

• Gamma distribution generalizes exponential distribution to model waiting time for multiple events
• PDF of gamma distribution $f(x) = \frac{λ^k x^{k-1} e^{-λx}}{Γ(k)}$ for x ≥ 0, where k represents shape parameter and λ represents rate parameter
• Gamma function $Γ(k) = \int_0^∞ t^{k-1} e^{-t} dt$ used in gamma distribution formula
• Exponential distribution special case of gamma distribution when shape parameter k = 1
• Expected value of gamma distribution $E(X) = \frac{k}{λ}$
• Variance of gamma distribution $Var(X) = \frac{k}{λ^2}$

Applications in Reliability and Waiting Time

• Reliability theory uses exponential distribution to model time until failure of electronic components
• Gamma distribution models time until k failures occur in a system
• Waiting time in queues often modeled using exponential distribution (time between arrivals)
• Service times in queueing theory may follow gamma distribution (sum of exponential service times)
• Insurance claims frequency modeled with exponential distribution, claim sizes with gamma distribution

Distribution Parameters

Rate and Shape Parameters

• Rate parameter λ in exponential distribution represents average number of events per unit time
• Inverse of rate parameter $\frac{1}{λ}$ gives mean waiting time between events
• Shape parameter k in gamma distribution determines distribution's shape
  • k = 1 yields exponential distribution
  • k < 1 creates more right-skewed distribution
  • k > 1 results in less skewed, more symmetric distribution
• Increasing shape parameter k in gamma distribution shifts probability mass towards right

Scale Parameter and Memoryless Property

• Scale parameter θ in gamma distribution related to rate parameter by $θ = \frac{1}{λ}$
• Scale parameter affects spread of distribution without changing its shape
• Memoryless property of exponential distribution states $P(X > s + t | X > s) = P(X > t)$ for all s, t ≥ 0
• Memoryless property implies no "aging" or "wear-out" in exponential model
• Gamma distribution does not possess memoryless property except when k = 1 (exponential case)

Distribution Functions and Properties

Probability Density and Cumulative Distribution Functions

• PDF of exponential distribution $f(x) = λe^{-λx}$ represents probability density at each point
• CDF of exponential distribution $F(x) = 1 - e^{-λx}$ gives probability of event occurring by time x
• PDF of gamma distribution $f(x) = \frac{λ^k x^{k-1} e^{-λx}}{Γ(k)}$ more complex due to shape parameter
• CDF of gamma distribution $F(x) = \frac{γ(k, λx)}{Γ(k)}$ where γ(k, λx) represents lower incomplete gamma function
• Both PDFs integrate to 1 over their support (0 to ∞)

Expected Value, Variance, and Other Properties

• Expected value of exponential distribution $E(X) = \frac{1}{λ}$ represents average waiting time
• Variance of exponential distribution $Var(X) = \frac{1}{λ^2}$ measures spread around mean
• Coefficient of variation for exponential distribution always equals 1
• Expected value of gamma distribution $E(X) = \frac{k}{λ}$ increases with shape parameter
• Variance of gamma distribution $Var(X) = \frac{k}{λ^2}$ also increases with shape parameter
• Mode of gamma distribution (k - 1)/λ for k > 1, 0 for k ≤ 1
• Moment generating function for exponential distribution $M_X(t) = \frac{λ}{λ-t}$ for t < λ