# Hazard Distribution

Let $Y$ be a random variable denoting the age at which a piece of equipment fails. In reliability theory, the probability that an item fails at time $y$ given that it has survived until time $y$ is called the hazard rate, $h(y)$. In terms of the pdf and cdf, $$h(y)= \frac{f_Y (y)}{1− F_Y(y)}.$$

Find $h(y)$ if $Y$ has an exponential pdf.

HINT Let $Y$ have exponential distribution. What is the pdf $f_Y(y)$ and the cdf $F_Y(y)$? The hazard rate is defined by $$h(y) = \frac{f_Y(y)}{1-F_Y(y)}.$$ Plug both into this definition and simplify.