Question

If the p.d.f of a c.r.v. X is
f(x) = `{:("K.e"^(-thetax)";", theta > 0","  0 ≤ x < oo),(0";", "otherwise"):}`
Then, K = ______.

Options

• `theta/3`

• `theta/2`

• θ

• 2θ

Answer 1

If the p.d.f of a c.r.v. X is
f(x) = `{:("K.e"^(-thetax)";", theta > 0","  0 ≤ x < oo),(0";", "otherwise"):}`
Then, K = θ.

Explanation:

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

∴  `int_-oo^oo "f"(x)  "d"x` = 1

∴ `int_0^oo "K.e"^(-thetax)  "d"x` = 1

∴ `"K"["e"^(-thetax)/(-theta)]_0^oo` = 1

⇒ `- "K"/theta [1/"e"^(thetax)]_0^oo` = 1

⇒ `- "K"/theta [1/"e"^oo - 1/"e"^0]` = 1

⇒ `- "K"/theta [1/oo - 1/1]` = 1

⇒ `"K"/theta` = 1

⇒ K = θ