Question

X is a continuous random variable with a probability density function

f(x) = `{{:(x^2/4 + k;     0 ≤ x ≤ 2),(0;              "otherwise"):}`

The value of k is equal to ______

Options

• `1/12`

• `1/3`

• `1/4`

• `1/6`

Answer 1

X is a continuous random variable with a probability density function

f(x) = `{{:(x^2/4 + k;     0 ≤ x ≤ 2),(0;              "otherwise"):}`

The value of k is equal to `underline(1/6)`.

Explanation:

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

∴ `int_{-∞}^∞ f(x)  dx = 1`

⇒ `int_0^2(x^2/4 + k) dx = 1`

⇒ `[x^3/12 + kx]_0^2 = 1 ⇒ 2/3 + 2k = 1`

⇒ `2k = 1/3 ⇒ k = 1/6`