# Standard Gamma Distribution

• kdizzle711
In summary, the conversation discusses an example in a book with omitted steps, involving a reaction time X of a randomly selected individual to a certain stimulus. The standard gamma distribution with alpha=2 is used, and the continuous variable X is evaluated using the equation F(x;alpha)=integral from 0 to x [(y^(alpha-1))*(e^-y)]/(gamma(alpha). The problem at hand is calculating P(2<=X<=5), with the result being .159. The person in the conversation is having trouble getting the same result, despite following the equation and assuming gamma(alpha)=1. They are seeking help for an upcoming test.

## Homework Statement

This is an example in my book with omitted steps. Suppose the reaction time X of a randomnly selected individual to a certain stimulus has a standard gamma distribution with alpha=2. When X is continuous

P(2<=X<=5) = F(5;2)-F(3;2) = .960-.801 = .159

## Homework Equations

F(x;alpha) = integral from x to 0 [(y^(alpha-1))*(e^-y)]/(gamma(alpha)

## The Attempt at a Solution

I have followed the equation and placed 5 for y and 2 for alpha but the numbers are not matching up. In the test it says that gamma(alpha) is equal to one. Please help I have a test tommorow. Thanks

What exactly is the problem? What numbers are you expecting and what numbers are you getting?

You write:
P(2 <= X <= 5) = .159
but is this what you're supposed to get, or what you're actually getting?

I get P(2 <= X <=5)=.159 as well using the exact formulas you posted (minor nitpick: the integral is from 0 to x, not from x to 0, but I assume this was just a typo).

## 1. What is a Standard Gamma Distribution?

A Standard Gamma Distribution is a continuous probability distribution that is commonly used to model the waiting time or the time between events. It is a two-parameter distribution that is defined by its shape and scale parameters.

## 2. How is a Standard Gamma Distribution different from a Normal Distribution?

A Standard Gamma Distribution differs from a Normal Distribution in several ways. Firstly, it is a continuous distribution while a Normal Distribution is a continuous distribution. Secondly, a Standard Gamma Distribution is right-skewed, while a Normal Distribution is symmetric. Lastly, a Standard Gamma Distribution has two parameters, while a Normal Distribution has only one.

## 3. What is the shape parameter in a Standard Gamma Distribution?

The shape parameter in a Standard Gamma Distribution, also known as the alpha parameter, determines the shape of the distribution curve. It represents the degree of skewness in the distribution, with higher values leading to a more right-skewed curve.

## 4. How are the parameters of a Standard Gamma Distribution related to its mean and variance?

The mean of a Standard Gamma Distribution is equal to the shape parameter divided by the scale parameter. The variance is equal to the shape parameter divided by the square of the scale parameter. In other words, the mean and variance of a Standard Gamma Distribution are directly proportional to the shape parameter, and inversely proportional to the scale parameter.

## 5. What are some real-world applications of the Standard Gamma Distribution?

The Standard Gamma Distribution has many real-world applications, including modeling the time between customer arrivals, the time between equipment failures, and the time between phone calls at a call center. It is also commonly used in the field of finance to model the time between stock price changes and the time between insurance claims. Additionally, the Standard Gamma Distribution is used in survival analysis to model the time until an event, such as death or recovery, occurs.

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