# Survival Analysis - Relationship between CDF and survival function

I am trying to teach myself survival analysis and I followed this youtube video.

The video states that given:

$$F_x(t) = \Pr[T_x \leq t ]$$ is the future probability of the life for a person who has reached $$x$$ years of age (aka c.d.f.)

and

$$S_x(t) = \Pr[T_x > t ] = 1 - F_x(t)$$ is the is the probability that a person of age $$x$$ will be alive for more than $$t$$ years (aka survival function).

The probability of someone $$x$$ years old dying in the next $$t$$ years is:

$$F_x(t) = \Pr[T_x \leq x + t | T_x > t] (Eq.1)$$

To me, it makes sense to think of this as the intersection of all those who will die at $$x+t$$ and who survived up until $$T_x>t$$. (1. Is this the correct intuition for this?)

The video goes on to state that we "rewrite the function (Eq.1) given that P(A) intersection P(B)" as the following:

$$F_x(t) = \frac{F(x + t)- F(x)}{P(T_x>x)} = F_x(t) = \frac{F(x + t)- F(x)}{S(x)} (Eq.2)$$

My understanding of $$P(A)$$ intersection $$P(B)$$ is that $$P(A\cap B) = P(A|B)P(B)$$

If, from $$Eq.1$$ if I let $$P(A) = T_x \leq x + t$$ and $$P(B) = T_x > t$$ then the resulting equation would be:

$$P(A\cap B) = P(A|B)P(B) = [(T_x \leq x + t)|(T_x > t)](T \leq x + t)$$

I feel like I am not doing this correctly. 2. How does $$S_x(t)$$ end up in the denominator in Eq.2?

I also have Survival Data: Extending the Cox Model by Therneau and Grambsch as a resource but they gloss over where the equations come from. I feel like this is leading up to the hazard function, but I want to make sure I understand what he going on before considering limits.

EDIT:

I don't think my question is clear enough. What I meant by "How does $$S_x(t)$$ end up in the denominator in Eq.2". I meant, how do we go from

$$F_x(t) = \Pr[T_x \leq x +t|T_x >t]$$ to $$\frac{F(x+t)-F(x)}{P(T_x>x)}$$

I am not sure how, algebraically, we get $$T_x >t$$ in the denominator

• $F_x(t) = \Pr[T_x \leq t ]$ vs $F_x(t) = \Pr[T_x \leq x + t | T_x > t]$ Commented Dec 1, 2018 at 14:42
• I believe I understand what you are saying - in my equations the subscript, $x$ in $T_x$ is misleading. Correct? Commented Dec 1, 2018 at 22:28
• I do not know what $F_x(t)$ is because you have two $F_x(t)$. Commented Dec 1, 2018 at 22:39

Derivation: For $$0 < x < x+t,$$ we have
$$F_x(t) = P(T \le x+t\,|\, T > x) = \frac{P(T \le x+t,\, T > x)}{P(T > x)} \\ = \frac{P(x x)} = \frac{F(x+t) - F(x)}{S(x)},$$ where $$F(y) = P(T\le y)$$ and $$S(y) = P(T > y) = 1 - F(y).$$
• I don't think my question is clear enough. What I meant by "How does $S_x(t)$ end up in the denominator in Eq.2". I meant, how do we go from $F_x(t) to \Pr[T_x \leq x +t|T_x >t] = \frac{F(x+t)-F(x)}{P(T_x>x)}$ Commented Dec 1, 2018 at 21:45
• The sentence "The probability of someone $x$ years old dying in the next $t$ years is: $F_x(t) = \Pr[T_x \leq x + t | T_x > t].$" makes no sense to me. The words suppose that someone is alive at time $x,$ so in my answer I changed the conditioning event to $\{T > x\}.$ That may not be exactly what you intended, but at least it makes some sense to me. Commented Dec 1, 2018 at 22:27
• I believe I understand what you are saying - in my equations the subscript, $x$ in $T_x$ is misleading. Correct? (Also, I don't mind you being blunt, I would rather get this right!) Commented Dec 1, 2018 at 22:29