Memoryless distribution

A real random variable $𝑋 : 𝜉 \to ℝ$ is said to be memoryless iff

ℙ (𝑋 \geq 𝑠 + 𝑡 ∣ 𝑋 \geq 𝑠) = ℙ (𝑋 \geq 𝑡)

or equivalently

ℙ (𝑋 \geq 𝑠) ℙ (𝑋 \geq 𝑡) = ℙ (𝑋 \geq 𝑠 + 𝑡)

for all $𝑠, 𝑡 > 0$ . A positive continuous random variable is memoryless iff it has the exponential distribution prob

Proof

proof