Relationship between sample mean and sample survival probability

[Blain Waan]

I am not sure if this question is very naive, but I was wondering if there is a relationship between sample mean and sample survival probability. For an exponential distribution with mean mu,
S(t)=1-exp(-t/mu})
Now if I generate data from an exponential distribution, can the sample mean $\bar{x}$ be used to find S.hat(t) as an estimate of S(t)?

How about if I generate lifetimes from exponential distribution with different means (say, for males and females). Can the sample mean be used to find out the sample survival probability as an estimate of population survival probability then too?

 

[BGM]

Sample mean \( \hat{\mu} \) is the MLE of the parameter \( \mu \). By the invariance property of the MLE,

\( \hat{S}(t) = 1 - \exp\left\{-\frac {t} {\hat{\mu}}\right\} \)

is the MLE of \( S(t) \).

That means it make sense to use \( \hat{S}(t) \) to estimate \( S(t) \).

Not sure what is the difference in your second part.

 

[Blain Waan]

Thank you. Say, if I generate lifetimes for males with exp(65yrs) and for females with exp(62yrs), then can I use the sample mean to find S.hat(t)?