Given that x is a random variable in range [0, ∞] with a probability density function e−x/2/K, the value of the constant K is ______ .
For p.d.f, ∫∞−∞f(x)dx=1
⇒ ∫∞0e−x2K dx=1
⇒ 1K⎛⎜
⎜
⎜
⎜⎝e−x2(−12)⎞⎟
⎟
⎟
⎟⎠∞0=1
⇒ −2K⎛⎝e−x2⎞⎠∞0=1
⇒ −2K(0−1)=1
⇒ K=2