The pdf for T∼Exponential(λ)T\sim \text{Exponential}(\lambda)T∼Exponential(λ) is f(t∣λ)=λe−λt⋅It>0f(t|\lambda)=\lambda e^{-\lambda t} \cdot \mathbb{I}_{t\gt 0} f(t∣λ)=λe−λt⋅It>0 Its cdf is F(t∣λ)=(1−e−λt)⋅It>0F(t|\lambda)=(1-e^{-\lambda t})\cdot \mathbb{I}_{t\gt 0} F(t∣λ)=(1−e−λt)⋅It>0 which implies the survivor function is S(t):=P(T>t)=1−F(t∣λ)=e−λtS(t):=\mathbb{P}(T\gt t)=1-F(t|\lambda)=e^{-\lambda t} S(t):=P(T>t)=1−F(t∣λ)=e−λt