Answer 1

Since, f is the p.d.f. of X,

` int_(-∞)^∞ f (x) dx` = 1

∴ ` int_(-∞)^-2 f (x) dx` +` int_(-2)^2 f (x) dx` + ` int_(2)^∞f (x) dx`= 1

∴ 0 + ` int_(-2)^2 k (4 -x^2) dx` = 1

∴ k ` int_(-2)^2  (4 -x^2) dx` = 1

∴ k` [ 4x - x^3/3]_-2^2` = 1

∴ k `[(8-8/3)-(-8+8/3)]`= 1

∴ k`(16/3+16/3)` = 1

∴ k`(32/3)` = 1

∴ k = `3/32`

P(x > 0)

= ` int_(0)^∞ f (x) dx`

= ` int_(0)^2 f (x) dx`+ ` int_(2)^∞ f (x) dx`

= ` int_(0)^2 k (4-x^2) dx+ 0`

= k` int_(0)^2  (4-x^2) dx`

=`3/32[4x -x^3/3]_0^2`  ..........[∵ k=`3/32`]

=`3/32 [8-8/3] = 3/32 xx16/3 = 1/2`