When a certain procedure consists of n𝑛 independent steps and each step takes Exponential(λ𝜆) amount of time then the total time for these n𝑛 steps will follow:a.Exponential distributionb.Gamma distributionc.Normal distributiond.Uniform distribution
Question
When a certain procedure consists of n𝑛 independent steps and each step takes Exponential(λ𝜆) amount of time then the total time for these n𝑛 steps will follow:a.Exponential distributionb.Gamma distributionc.Normal distributiond.Uniform distribution
Solution
The total time for these n independent steps will follow a Gamma distribution.
Here's why:
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The exponential distribution is often used to model the time between independent events that happen at a constant average rate. In this case, each step of the procedure is an independent event that takes an Exponential(λ) amount of time.
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When you add up n independent Exponential(λ) random variables (i.e., the time it takes to complete each step), the sum follows a Gamma distribution. This is a property of the exponential and gamma distributions.
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The Gamma distribution has two parameters: shape and scale. In this case, the shape parameter would be n (the number of steps), and the scale parameter would be 1/λ (the average time per step).
So, the total time for the n steps follows a Gamma(n, 1/λ) distribution.
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