Multiple independent exponential random variables

In summary, Multiple independent exponential random variables are a set of random variables that are independent from each other and follow the exponential distribution. They are commonly used in statistical analysis and modeling in various fields such as biology, physics, and economics. They differ from other types of random variables in terms of their distribution and independence. The probability of multiple independent exponential random variables can be calculated using a specific formula. Some real-life examples of these variables include the time between radioactive decay events, incoming phone calls, and earthquakes. They can also be used to model the lifetime of electronic components or the time between failures in a machine.
  • #1
libragirl79
31
0
Let X1, X2, ...Xn be independent exponential variables having a common parameter gamma. Determine the distribution of min(X1,X2, ...Xn).




The Attempt at a Solution


I know how to do it with one X and one parameter but I am at a loss with these multiple ones...

Thanks so much!
 
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  • #2
Hint: min(X1,X2,...Xn)>x is equivalent to saying X1>x and X2>x and... Xn>x.

You can find the latter probability by multiplying the probabilities of each event.
 

FAQ: Multiple independent exponential random variables

1. What are multiple independent exponential random variables?

Multiple independent exponential random variables are a set of random variables that follow the exponential distribution and are independent from each other. This means that the outcome of one variable does not affect the outcome of the others.

2. How are multiple independent exponential random variables used in science?

Multiple independent exponential random variables are commonly used in statistical analysis and modeling. They are also used in various fields such as biology, physics, and economics to represent the time between events that occur randomly and independently.

3. How are multiple independent exponential random variables different from other types of random variables?

Multiple independent exponential random variables are different from other types of random variables, such as normal or binomial, because they follow the exponential distribution which has a specific shape and characteristics. Additionally, they are independent from each other, while other types of random variables may be dependent or correlated.

4. What is the formula for calculating the probability of multiple independent exponential random variables?

The probability of multiple independent exponential random variables can be calculated using the formula P(X1>x1, X2>x2, ..., Xn>xn) = e^(-λ1x1) * e^(-λ2x2) * ... * e^(-λnxn) = e^(-(λ1x1 + λ2x2 + ... + λnxn)), where λ is the rate parameter and x is the time interval.

5. What are some real-life examples of multiple independent exponential random variables?

Real-life examples of multiple independent exponential random variables include the time between radioactive decay events, the time between incoming phone calls at a call center, and the time between earthquakes in a specific region. They can also be used to model the lifetime of electronic components or the time between failures in a machine.

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