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The Gamma_Dist command visualizes the Gamma distribution through probability and frequency-domain charts. The Gamma distribution is a flexible continuous probability distribution commonly used to model waiting times, lifetimes, insurance losses, and financial risk measures.
Main idea: model the time required for multiple random events to occur and analyze positively skewed continuous data.
Gamma_Dist(2,1)
This example creates a Gamma distribution with shape parameter 2 and scale parameter 1.
Gamma_Dist(shape,scale)
shape – shape parameter (α or k)scale – scale parameter (θ)
f(x) =
x^(α-1)e^(-x/θ)
-------------------
Γ(α)θ^α
for x ≥ 0.
The mean and variance are:
Mean = αθ
Variance = αθ²
The Gamma distribution generalizes the Exponential distribution. While the Exponential distribution models the waiting time until a single random event occurs, the Gamma distribution models the waiting time until multiple events occur.
As the shape parameter increases, the distribution becomes less skewed and gradually approaches a Normal distribution.
The Gamma distribution appears frequently in quantitative finance. For example, waiting times between market events, transaction arrivals, and certain stochastic volatility models can be represented using Gamma-related processes.
The Characteristic Function panels provide a frequency-domain representation of the distribution. Characteristic functions uniquely identify probability distributions and play an important role in Fourier methods, option pricing, signal processing, and advanced statistical analysis.