Math Problem Statement
If a random variable x is Gamma distribution with parameter λ = 3, compute the value of
- P(x ≤ 1) 2. P(1 ≤ x ≤ 2)
Solution
Given that is a random variable that follows a Gamma distribution with parameter , we need to compute the following probabilities:
Let's proceed step-by-step:
Gamma Distribution
The Gamma distribution's probability density function (PDF) for a random variable is given by:
where:
- is the shape parameter,
- is the rate parameter,
- is the Gamma function, which generalizes the factorial function.
Interpreting the Problem
The problem provides us with , but we do not have an explicit value for (shape parameter). If only is given and no is specified, a typical interpretation (or the most straightforward) would be to assume the standard exponential distribution, where . This is because the exponential distribution is a special case of the Gamma distribution with .
Hence, we assume:
- Shape parameter
- Rate parameter
Exponential Distribution PDF
With , the Gamma distribution simplifies to an exponential distribution with PDF:
For our problem, this becomes:
Computing the Probabilities
- Probability
Using integration by substitution, let , hence , which gives . The integral becomes:
- Probability
Following a similar method:
Final Answers
Would you like more details, or do you have any questions?
Related Questions
- How does the Gamma distribution differ from the exponential distribution?
- What is the cumulative distribution function (CDF) of the exponential distribution?
- How can we interpret the shape and rate parameters of the Gamma distribution?
- How would the probability change if the shape parameter were different?
- What are the mean and variance of a Gamma-distributed random variable?
Tip
The exponential distribution is a special case of the Gamma distribution and is commonly used to model the time until the next event in a Poisson process.
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Math Problem Analysis
Mathematical Concepts
Gamma Distribution
Exponential Distribution
Probability Theory
Formulas
Gamma distribution PDF
Exponential distribution PDF
Integral calculus
Theorems
-
Suitable Grade Level
Advanced Mathematics
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