Please I need help on this question

Question

Let X_1,X_2,…,X_n be independent random variables from a symmetrically distributed function over (0, 1). Let y_i be the order statistics. Find the probability that; 0.3<y_3<0.7; if n=5.

Answer 1

 

The distribution is centered at x=1/2

 

line with slope 2 begins at (0,0) and goes to (1/2,1)

 

and another line with slope -1/2 goes from (1,0) to (1/2,1)

 

The area of this region is 1, so it satisfies the symmetric distribution

 

We are interested in the area between x=0.3 and x=0.7

 

Dividing this area into a triangle by the horizontal line segment from (0.3,0.6) to (0.7,0.6)

 

the area of the rectangular portion is 0.4 * 0.6 = 0.24 

the area of the triangular  portion is 1/2*0.4*0.4 = 0.08

 

The area, hence the probability is 0.32