Question

Find k if the following function represent p.d.f. of r.v. X

f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P `(1/ 4 < x < 3 /2)`.

Answer 1

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

∴` int_(-∞)^∞ f (x) dx` = 1

∴` int_(-∞)^0 f (x) dx`+ ` int_(0)^2 f (x) dx` + ` int_(2)^∞ f (x) dx`  = 1

∴ 0 + ` int_(0)^2 kxdx` + 0 =1

∴ `k [x^2/2]_0^2` =1

∴ `k [4/2 - 0] =1`

∴ 2k = 1

∴ k =`1/2`

P `(1/ 4 < x < 3 /2)` = ` int_(1/4)^(3/2) f (x) dx`

` int_(1/4)^(3/2) kxdx`,

where k= `1/2`

= `1/2  int_(1/4)^(3/2)  x dx`

= `1/2[x^2/2]_(1/4)^(3/2)`

= `1/4[9/4-1/16]`

= `1/4[(36-1)/16]`

=`35/64`