指数分布

The exponential distribution is a continuous probability distribution that is often used to model the time until an event occurs, such as the time between arrivals of customers at a service center, the life of a machine before it breaks down, or the time until a radioactive particle decays. It is a particular case of the gamma distribution with a specific parameterization.

Here are some key characteristics and properties of the exponential distribution:

1. **Memorylessness:** The exponential distribution has the memoryless property, which means that the probability of an event occurring in the next moment is not affected by how much time has already elapsed. In other words, the future life expectancy is independent of the past.

2. **Parameter:** The exponential distribution is characterized by a single parameter \( \lambda \) (lambda), which is the rate parameter. It represents the average number of events in a given time interval.

3. **Probability Density Function (PDF):** The PDF of an exponential distribution is given by:
\[ f(x;\lambda) = \lambda e^{-\lambda x} \]
for \( x \geq 0 \), and \( f(x;\lambda) = 0 \) for \( x < 0 \).

4. **Cumulative Distribution Function (CDF):** The CDF, which gives the probability that a random variable \( X \) is less than or equal to a certain value \( x \), is:
\[ F(x;\lambda) = 1 - e^{-\lambda x} \]
for \( x \geq 0 \).

5. **Mean and Variance:** The mean (expected value) of an exponential distribution is \( \frac{1}{\lambda} \), and the variance is \( \frac{1}{\lambda^2} \).

6. **Relationship to Poisson Process:** The exponential distribution is closely related to the Poisson process, where it describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

The exponential distribution is often used in reliability engineering and queuing theory due to its simplicity and its property of being memoryless, which makes it suitable to model systems without aging properties (where the failure rate is constant over time).