Discrete survivor function expressed in terms of hazard Let $T$ can take on values $t_1,t_2,\ldots,$ with $0\le t_1\le t_2,\ldots,$ and let the probability function be
$$f(t_j)=Pr(T=t_j),\quad j=1,2,\ldots$$
The survivor function is then 
$$S(t)=Pr(T\ge t)=\sum_{j:t_j\ge t}f(t_j)$$
The discrete time hazard function is defined as :
$$h(t_j)=\frac{f(t_j)}{S(t_j)}=1-\frac{S(t_j)}{S(t_{j+1})},\quad j=1,2,\ldots$$
since  $f(t_j)=S(t_j)-S(t_{j+1})$.


*

*But i can't able to derive : $S(t)=\prod_{j:t_j\le t}[1-h(t_j)]$

 A: Using the discrete nature of $T$, here is a heuristic way to derive the survival function at time $t_j$:
$$
\begin{align*}S(t_j) & = \text{surviving past time }t_j \\ 
 & = \text{surviving past time } t_{j-1} \text{ and not dying at } t_j \\
 & = \Pr\big(\text{not dying at }t_j \,\big|\, \text{surviving past } t_{j-1}\big) \times \Pr\big(\text{surviving past time } t_{j-1}\big) \\
 & = \Pr\big(\text{not dying at }t_j \,\big|\, \text{survival up to } t_j\big) \times \Pr\big(\text{surviving past time } t_{j-1}\big) \\
 & = [1 - h(t_j)] \times S(t_{j - 1})
\end{align*}$$ 
A: For all $j=1,2,\ldots$ we have
$$
1-h(t_j)=\frac{P(T\geq t_j)-P(T=t_j)}{P(T\geq t_j)}=\frac{P(T\geq t_{j+1})}{P(T\geq t_j)}.
$$
Now, if $t>0$, then we choose the largest $t_k$ such that $t_k\leq t$. Thus
$$
\prod_{j:t_j\leq t}[1-h(t_j)]=\prod_{j:t_j\leq t_k}[1-h_j(t)]=\prod_{j=1}^k\frac{P(T\geq t_{j+1})}{P(T\geq t_j)}.
$$
Noting that this is a telescoping product, we end up with
$$
\prod_{j:t_j\leq t}[1-h(t_j)]=\frac{P(T\geq t_{k+1})}{P(T\geq t_1)}=P(T>t_k)=S(t_k)=S(t).
$$
