Solving Order Statistics with Three Uniformly Distributed Random Variables

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In summary, the problem asks to find the probability that three random variables, X1, X2, and X3, generated on a spinning fair wheel three times, are all within +-d of each other. These variables are independent and uniformly distributed on [0,1]. The question also asks about the probability when considering the order statistics for the variables, where 0<=Y1<=Y2<=Y3<=1. To solve this, it is necessary to consider different cases based on the variables falling within d of the endpoints or within 2d of each other, and whether the values fall on an interval or a circle.
  • #1
robert5
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Three random variables are generated X1, X2, X3 on a spnning fair wheel three times. these variables are independent and uniformaly distributes on [0,1]. find probability that these values are none within +-d of each other where 0<=Y1<=Y2<=Y3<=1 is order statistics for randon variables.
fY2Y3(y2,y3) = 2!fx(y) . fX(y) =

what is fX(y)? can some one help?

Also,

Pr[d<=Y2<=(1.2d), (y2+d)<=Y3<=(1-d)] =

where y2 and y3 be placed in [0, 1]

I can understand it but don't know how to do it...
 
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  • #2
Hi Guys,

Any help ?
 
  • #3
I'm interested in knowing how to do this, too. If the problem is asking what I think it is, it seems that it could be generalized to 2-dimensions. That might provide a simplified model of the number of clusters of balls remaining after the break shot in a game of pool (pocket billiards)...
 
  • #4
robert5 said:
Three random variables are generated X1, X2, X3 on a spnning fair wheel three times. these variables are independent and uniformaly distributes on [0,1]. find probability that these values are none within +-d of each other where 0<=Y1<=Y2<=Y3<=1 is order statistics for randon variables.

You don't need to the order statistics to solve this, however you will need to consider several separate cases where x1 or x2 fall within d of the endpoints or within 2d of each other. Also do the values fall on an interval or on a circle. The latter case will be a bit simpler to solve.
 
  • #5
thanks your solving problems
 

FAQ: Solving Order Statistics with Three Uniformly Distributed Random Variables

1. What is the purpose of solving order statistics with three uniformly distributed random variables?

The purpose of solving order statistics with three uniformly distributed random variables is to understand the probability distribution of the k-th order statistic in a set of three independent and identically distributed random variables with a uniform distribution.

2. What are the steps involved in solving order statistics with three uniformly distributed random variables?

The steps involved in solving order statistics with three uniformly distributed random variables include finding the probability density function, calculating the cumulative distribution function, and determining the expected value and variance of the k-th order statistic.

3. What is the significance of the k-th order statistic in solving order statistics with three uniformly distributed random variables?

The k-th order statistic represents the k-th smallest value in a set of three random variables, and it is used to calculate the probability of obtaining a value within a certain range or above a certain threshold in the set.

4. How does the number of uniformly distributed random variables affect the solution for order statistics?

The number of uniformly distributed random variables affects the solution for order statistics by changing the probability density function and the expected value and variance of the k-th order statistic. As the number of random variables increases, the probability of obtaining extreme values decreases.

5. Are there any real-world applications of solving order statistics with three uniformly distributed random variables?

Yes, there are many real-world applications of solving order statistics with three uniformly distributed random variables, such as predicting the likelihood of a team winning a sports game based on their past performances, determining the probability of a stock reaching a certain price based on its historical data, and estimating the number of customers who will purchase a product based on their previous buying behaviors.

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