Suppose that X is a uniformly distributed random variable ov

Suppose that X is a uniformly distributed random variable over [-a,a] where a>0. Whenever possible, determine the value of a so that the following are satisfied: P(|X|<1)=P(|X|>1)

Solution

pdf= 1 / ( a- (-a)) = 1/ 2a when -1 < x < +1..........
   = 0 otherwise...........
P(|X|<1)=P(|X|>1) i.e , P(|X|<1) = 1- P(|X|<1) .... i.e, 2* P(|X|<1) = 1. i.e, P(|X|<1)= 0.5

i.e, p(-1 <X < 1) = 0.5 . i.e, p(x<1) -p(x< -1) = 0.5
i.e, integration from -a to 1( dx/2a) - integration from -a to -1 ( dx/2a ) = 0.5

i.e, [ (1+a)/2a - (-1 + a ) / 2a ] = 0.5 or, 1/a = 0.5 or, a= 2