1. Support
In probability theory, "support" (support set) refers to the set of possible values for a random variable. For discrete random variables, the support set contains all possible values; for continuous random variables, the support set refers to the area where its density function or probability mass function is non-zero.
For example, for a discrete random variable, such as the result of a coin toss (heads or tails), the support set is {heads, tails}, because these two are the only possible outcomes.
For a continuous random variable, such as a random variable obeying the standard normal distribution, its support set is the entire real number axis, because the probability density function is defined and non-zero on the entire real number axis.
The concept of a support set is important in probability theory because it determines the value range and possibility of a random variable.
2. hazard rate function
In probability theory, "hazard rate" (hazard rate) refers to the probability density of an event occurring in the next time interval, given the conditions in which an event occurs at a given point in time. Risk rate is often used to describe the change in the probability of failure or failure of a random event over time.
For a random event, its hazard rate function can be expressed as h(t), where t represents time. The hazard rate function tells us the rate or probability density of event failure at a specific point in time.
Specifically, for a continuous random variable X, its hazard rate function can be defined as:
h(t) = f(t) / S(t)
Among them, f(t) is the probability density function of the event failure at time point t, and S(t) is the cumulative distribution function.
Hazard rates are very useful in reliability engineering and survival analysis, and can help researchers estimate and analyze the probability and time correlation of random events. By analyzing risk rates, one can assess the reliability of events, lifetime distributions, and predict future risk situations.