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# Category Archives: Uniform

## Uniform Spacing

Let \(U_1, U_2, U_3, U_4, U_5\) be independent uniform \((0,1)\) random variables. Let \(R\) be the difference between the max and the min of the random variables. Find

- \( E( R)\)
- the joint density of the min and the max of the \(U\)’s
- \(P( R>0.5)\)

[Pitman p. 355 #14]

## Mean and Mode of Binomial

Consider a binomial\((10,p)\) distribution. If \(p\) is chosen uniformly at random from the interval \((0,1)\), what is the likelihood that the most likely number of the binomial distribution will be less than the mean of the binomial distribution?