Are Ratios of IID Exponential Variables Independent of Their Sample Average?

In summary, the conversation discusses the independence of the ratio of two independently, identically distributed exponential random variables and the sample average. It is noted that this ratio has a Pareto distribution and the reasoning behind its independence from the sample average is attributed to scale invariance. However, it is also mentioned that X/Y and X+Y are not always independent.
  • #1
e12514
30
0
Suppose I have a sample X_1, ..., X_n of independently, identically distributed exponential random variables.

One result I deducted was that the ratio of any two of them (eg. X_1 / X_2) is independent of the sample average 1/n * \sum_{i=1}^{n} X_i.
(Aside: that ratio, as a random variable, has a Pareto distribution)

What's the reasoning/ intuitive appeal behind that? I know that any datapoint from an independently, identically distributed sample is in general not independent of the sample average unless there is zero variance, so.. How do we interpret this result here? Why is the ratio independent of the sample average?
 
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  • #2
My gut feeling is that this is a manifestation of some sort of scale invariance. Incidentally, we can simplify the situation by doing away with all of the other variables -- we're just looking at the independence of X/Y and X+Y.
 
  • #3
Are X/Y and X+Y independent (given X and Y are)? I can't seem to show that in general...
 
  • #4
e12514 said:
Are X/Y and X+Y independent (given X and Y are)? I can't seem to show that in general...

No. Just to pick a simple example, suppose that X and Y are IID taking the values 1,2 each with a 50% probability.

X/Y=1/2 or X/Y = 2/1 <=> X+Y = 3
X/Y = 1 <=> X+Y = 2 or 4

so they aren't independent.
 

FAQ: Are Ratios of IID Exponential Variables Independent of Their Sample Average?

What is an exponential random variable?

An exponential random variable is a type of continuous probability distribution that is often used to model time between events. It is a non-negative variable, meaning that it can only take on values greater than or equal to zero. It is also a memoryless distribution, meaning that the probability of an event occurring in the next time interval is not affected by how much time has passed since the last event.

How is an exponential random variable different from other types of random variables?

Unlike other types of random variables, such as the normal or binomial distribution, an exponential random variable only has one parameter, called the rate parameter. This parameter determines the shape of the distribution and can be used to calculate probabilities and expected values. Additionally, an exponential random variable is a continuous distribution, meaning that it can take on any value within a certain range, rather than just specific values.

What is the formula for the probability density function of an exponential random variable?

The probability density function (PDF) of an exponential random variable can be expressed as f(x) = λe^(-λx), where λ is the rate parameter and x is the value of the random variable. This formula can be used to calculate the probability of a given value occurring, or to create a graph of the distribution.

How is an exponential random variable used in real-world applications?

An exponential random variable can be used to model a variety of real-world scenarios, such as the time between customer arrivals at a store, the time between failures of a machine, or the lifespan of a product. It is also commonly used in queuing theory and reliability analysis. In finance, it is used to model the time between price changes in financial markets.

How is an exponential random variable related to the Poisson distribution?

An exponential random variable and the Poisson distribution are closely related, as they are both used to model the time between events. In fact, if the events in a Poisson process occur at a constant rate, then the time between events can be modeled using an exponential random variable. Additionally, the sum of n independent exponential random variables with the same rate parameter follows a gamma distribution, which can be used to approximate a Poisson distribution.

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