What Is the Distribution of \( n \times \min(U_1, \dots, U_n) \)?

In summary, the "Distribution of n*min(u1,u2)" is a probability distribution used to model the minimum of two independent random variables, u1 and u2, multiplied by a constant n. It is important in various fields and can be calculated by finding the distribution of the minimum of the two random variables and multiplying it by n. It is different from the "Distribution of u1+u2" as it models the smallest value rather than the sum of values. This distribution can also be extended to more than two random variables, known as the "Distribution of n*mink(ui)".
  • #1
infk
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Homework Statement


Let [itex]{U_k}_{k \in \mathbb{N}}[/itex] be i.i.d from the uniform (0,1) distribution.
I need a formula for the cumulative distribution function of [itex]X_n[/itex], defined as
[itex]X_n := n* \min(U_1, \ldots ,U_n)[/itex]

Also some advice for [itex]X_n := \sqrt{n}* \min(U_1, \ldots ,U_n)[/itex] would be appreciated.

[itex]*[/itex] is meant to be multiplication..

Homework Equations


The Attempt at a Solution


Know already that [itex]P(min(U_1, \ldots ,U_n) \leq x) = 1 - (1-x)^n[/itex]
 
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  • #2
Sorry to bump this thread,but I still haven't figured it out. Does anyone know how to find it?
 

FAQ: What Is the Distribution of \( n \times \min(U_1, \dots, U_n) \)?

What is the "Distribution of n*min(u1,u2)"?

The "Distribution of n*min(u1,u2)" is a probability distribution used to model the minimum of two independent random variables, u1 and u2, multiplied by a constant n. It is often used in statistics and probability theory to describe the smallest of two values.

What is the significance of this distribution?

This distribution is important in various fields such as economics, engineering, and medicine. It is used to model scenarios where the smallest of two values is of interest, such as in inventory management, reliability analysis, or survival analysis.

How is the "Distribution of n*min(u1,u2)" calculated?

The distribution can be calculated by first finding the distribution of the minimum of the two random variables, u1 and u2, and then multiplying it by the constant n. The resulting distribution will have a different shape depending on the distributions of u1 and u2.

What is the difference between "Distribution of n*min(u1,u2)" and "Distribution of u1+u2"?

The "Distribution of n*min(u1,u2)" is used to model the smallest of two values, while the "Distribution of u1+u2" is used to model the sum of two values. The former is useful in scenarios where the smallest value is of interest, while the latter is useful in scenarios where the combined value is of interest.

Can the "Distribution of n*min(u1,u2)" be used for more than two random variables?

Yes, this distribution can be extended to more than two random variables. It is commonly known as the "Distribution of n*mink(ui)" and is used to model the minimum of k independent random variables, where k is a positive integer.

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