Could you please help me to find a survival function for Poisson distribution? I'm very new to this topic. Poisson distribution probability mass function:
$$f(k) = P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}.$$
So, I found a general definition of survival function:
$$S(t) = P(T > t) = \int_t^\infty f(u)\,du = 1 - F(t).$$
But as I understand, this one suits only continuous distributions and Poisson is a discrete one. Then I found this definition, which seems to suit a discrete case:
$$S(t_j) = S_j = P(T \geq t_j) = \sum_{k=j}^\infty f(k).$$
But as I wasn't counting it earlier, I'm not sure how to apply it. I need it for my university project. Also, found that this is a survivor function for Poisson:
But don't know how to find it.