Question

Note! : I need to know why all the answers except (e) are wrong. So you need to explain why e is the right answer, and (a-d) are not. Why are they asking for the fourth largest variable, how does this have an effect on whether its one answer or another? What if they were asking for the 1st largest, 5, largest, etc.? Why is the answer (e) right and (a) is not? I need answers to these questions for a like, please use https://www.wolframalpha.com/ for links to the graphs of the different distributions, you can just add them in the search bar for plots. Thanks..!

Let
X__i;, i = 1, 2 ,. . . , 7, be independent continuous stochastic variables evenly distributed over the unit interval] 0; 1 [.


The fourth largest of the seven variables follows a...?

a. Beta (3,3) distribution

b. A normal ( 1/2 , 1/84) distribution

c. Binomial (7, 1/2) distribution

d. Uniform (0,1) distribution

e. Beta (4,4 ) distribution

Answer 1

You want to know the distribution of kth order statistic of uniform distribution

for uniform distribution

f(x) = 1

F(x) = x

here n = 7 , k = 4 {fourth largest}

hence this is beta(4,7-4+1)

= Beta(4,4)

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Note that pdf of beta() distribution is