指数分布计算器
计算指数分布的概率和期望等待时间。
PDF f(x)
0.11156508
PDF f(x) vs Rate Parameter (lambda)
公式
How to Calculate Exponential Distribution Probabilities
Formulas
PDF: f(x) = lambda * e^(-lambda*x) for x >= 0
CDF: P(X <= x) = 1 - e^(-lambda*x)
The exponential distribution models the time between events in a Poisson process. The parameter lambda is the rate (events per unit time). The mean waiting time is 1/lambda. It has the memoryless property: P(X > s+t | X > s) = P(X > t).
计算示例
Buses arrive at a rate of 0.5 per minute. What is the probability of waiting at most 3 minutes?
- 01lambda = 0.5, x = 3
- 02PDF: f(3) = 0.5 * e^(-0.5*3) = 0.5 * e^(-1.5) = 0.5 * 0.2231 = 0.1116
- 03CDF: P(X <= 3) = 1 - e^(-1.5) = 1 - 0.2231 = 0.7769
- 04P(wait > 3 min) = e^(-1.5) = 0.2231
- 05Mean waiting time = 1 / 0.5 = 2 minutes
常见问题
What is the memoryless property?
The exponential distribution is the only continuous distribution with the memoryless property: the probability of waiting at least t more units does not depend on how long you have already waited. This makes it suitable for modeling systems with constant failure rates.
How does the exponential relate to the Poisson distribution?
If events follow a Poisson process with rate lambda, the time between consecutive events follows an exponential distribution with the same rate lambda. They are complementary views of the same process.
What is the variance of the exponential distribution?
The variance is 1/lambda^2. For example, with lambda = 0.5, variance = 1/0.25 = 4, and standard deviation = 2.
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