# How can I use a Cox proportional hazards model to obtain probabilities?

I am using a Cox proportional hazards model to estimate the probability of an event (failure of a mechanical device, say) occurring at discrete, finite times: $t = 0, 1, 2, \ldots$. In Cox's original paper, Regression Models and Life Tables, he says that, if you let $T$ be the random variable representing the time to failure of a given member of your population, and $T$ is discrete, then $$\lambda(t) = Pr(T = t|T \geq t).$$ So it seems to me that, if I estimate $\lambda_0(t)$ in the canonical Cox model formula $$\lambda(t) = \lambda_0(t) \exp \left( \sum \beta_i X_i \right)$$ as a "baseline" probability that failure occurs (that is, a sort of average of the probability of the event occurring at time $t$ when the $X_i$ are all zero), I can obtain an estimate of the probability of the event occurring at a time $t$ in the future for a device with risk characteristics $X_i$ by just plugging in the values of $X_i$ and the value of $t$ into the formula. Is this the correct way to use a Cox model, or am I not thinking hard enough?

First of all, hazards modeling is concerned with the hazard function. If $T$ is a continuous random variable representing failure time of a subject, then the hazard function is defined as follows: $$\label{first} \lambda(t)=\lim_{\Delta t \downarrow 0} \frac{\mathbb P(t \leq T<t+\Delta t | T \geq t)}{\Delta t}$$
In a Cox model done the right way, if we want to get the probability density function $f_T(t)$, it can be found by taking the product $$f_T(t) = \lambda(t)S(t),$$ where $S(t)$ is the survival function, given by $$S(t) = \exp \left( -\int_0^t \lambda(s)ds \right).$$ What I don't really understand is how, precisely, these three equations need to be modified if $T$ is discrete. In Cox's paper, he says that the hazard function for discrete $T$ is (contrary to what I said in my original question) $$\lambda (t) = \sum_{j = 1}^N \mathbb P(T = t_j|T\geq t_j)\delta(t-t_j),$$ where $\{t_j\}_{j=1}^N$ is the set of points where $T$ has mass. Is there some way to obtain this from the first equation above?