# What is the hazard-rate of a truncated probability distribution?

Suppose $$X$$ is a random variable with pdf and cdf in forms of $$f(X)$$ and $$F(X)$$, with hazard-rate $$h(X)$$. Now, we define a new variable $$Y$$ which is a truncated random variable, $$Y=X, \quad \text{if } X \le a$$, and $$Y=0, \quad \text{if } X > a$$. I am wondering what is the hazard-rate of the random variable $$Y$$? For example, if $$X$$ is a geometric random, what is the hazard-rate of $$Y$$?

• Since the cdf of $Y$ has jump discontinuities, I think the first question is whether the hazard function for it is even well-defined.
– prubin
Dec 6, 2021 at 22:11
• Dec 13, 2021 at 8:34

The blog entry Defining Hazard Rate at a Point Mass. "Applied Probability and Statistics in Actuarial Science and Financial Economics", contains what is needed to answer the question.

Define the Hazard Rate Function $$h_T(t) = \frac {P(T = t)}{P(T \ge t)}$$, at discrete mass points t.

First I will assume $$X$$ is a nonnegative continuous variable, as implied by mention of pdf of $$X$$. I will also assume $$a$$ is positive.

The question seems to make more sense if $$Y = a$$ when $$X \ge a$$, i.e., $$Y = \text{min}(X,a)$$. In that case, there is a discrete mass point of $$Y$$ at $$a$$, and the hazard rate function of $$Y$$ is $$1$$ at $$a$$, and the usual hazard rate applies for $$X < a$$.

If instead, $$Y = 0$$ when $$X \ge a$$, then there is a discrete mass point of $$Y$$ at $$0$$, and the hazard rate function of $$Y$$ is $$P(X \ge a)$$ at $$0$$ (because the denominator of the hazard rate function is $$1$$); and the usual hazard rate applies for $$X < a$$.

Now, in order to handle your question about $$X$$ being Geometric, assume that $$X$$ is a discrete nonnegative variable, and $$a$$ is a positive integer.

If $$X$$ is Geometric with parameter $$p$$, the hazard rate function of $$Y$$ is $$p$$ for all nonnegative integers, except at $$a$$. If $$Y = \text{min}(X,a)$$, the hazard rate function of $$Y$$ is $$1$$ at $$Y = a$$. If, instead, $$Y = 0$$ when $$X \ge a$$, the hazard rate function of $$Y$$ at $$0$$ is $$p + (1-p)^a$$.