# Why does survival probabilities not agree?

I'm working with some survival probabilities, and i've gotten kind of confused.

For simplicity, the death rate ( mu(x) ), where x is age in years, are considered constant in intervals of 1 year. In my assignment, the probability of surviving 1 year at age x is given as:

p(x,1) = e^(-integral[0,1] mu(x,1) ds). Since mu is constant in intervals of a year, then just e^( -mu(x,1) ).

But obviously, since mu(x,1) is constant in this one year, then the probability of surviving should also be 1 - mu(x,1)? But 1-x <= e^(-x).

What is the flaw in my calculation here?

You say the death rate is $$\mu(x)$$ where $$x$$ is age in years and that $$\mu(x)$$ is constant intervals of years. If we modify this statement and say that $$\mu(x)$$ is the probability density of death, then you are correct that $$\mu(x)$$ is also the 1-year death probability for a subject with age $$x$$. It would not be true that a survival probability is $$\exp(-x)$$. Note: a probability density is akin to the height of a normal "bell curve", and the area under the curve is considered a probability.
Another type of density that characterizes survival is the exponential with rate parameter $$\lambda$$. Exponential RVs are unique because they have a constant rate unlike Weibull. For exponentially distributed failure times, and if $$\lambda=1$$ then the survival probability beyond time $$x$$ is $$\exp(-x)$$.
One scenario that might characterize your confusion is if a survival regression model is used to model the $$\lambda$$ as a function of one or more regressors. In this case, you might use $$\mu(x)$$ to represent the model for the rate or intensity of the survival process, and age is just such an $$x$$ in the model, the 1 year survival probability for a subject with age $$x$$ would be $$\exp(-\mu(x))$$.