Question

f(x) =  `{:{(x^3/k;       0 ≤ x ≤ 2), (0;     "otherwise"):}` is a p.d.f. of X. The value of k is ______

Options

• `1/4`

• `3/4`

• 4

• 2

Answer 1

f(x) =  `{:{(x^3/k;       0 ≤ x ≤ 2), (0;     "otherwise"):}` is a p.d.f. of X. The value of k is 4. 

Explanation:

Since f(x) is the p.d.f. of X,

`int_-∞^∞ f(x) dx = 1`

⇒ `int_0^2 x^3/k dx = 1 ⇒ 1/k[x^4/4]_0^2 = 1`

⇒ `1/k[16/4] = 1 ⇒ k = 4`