Exponential distribution song

by Kurious Fox

The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is often used to model waiting times or the time until an event occurs. Here are some key properties and characteristics of the exponential distribution:

Probability Density Function (PDF)

The PDF of an exponential distribution is given by:
f(x; \lambda) = \lambda e^{-\lambda x}
for x \geq 0 , where \lambda > 0 is the rate parameter.

Cumulative Distribution Function (CDF)

The CDF of an exponential distribution is given by:
F(x; \lambda) = 1 - e^{-\lambda x}
for x \geq 0 .

Mean and Variance

The mean (expected value) of an exponential distribution is:
\text{E}(X) = \frac{1}{\lambda}

The variance of an exponential distribution is:
\text{Var}(X) = \frac{1}{\lambda^2}

Memoryless Property

The exponential distribution has the memoryless property, which means that the probability of an event occurring in the next t units of time is independent of how much time has already elapsed. Formally, for X being exponentially distributed,
P(X > s + t \mid X > s) = P(X > t)

Applications

The exponential distribution is widely used in various fields such as:

  • Queueing Theory: To model the time between arrivals of customers or services.
  • Reliability Engineering: To model the lifespan of electronic components or systems.
  • Survival Analysis: To model the time until an event such as failure or death.

Example

Suppose the average rate of receiving a call at a call center is 2 calls per minute (\lambda = 2). The time between calls can be modeled using an exponential distribution with \lambda = 2.

  • PDF: f(x) = 2 e^{-2x}
  • CDF: F(x) = 1 - e^{-2x}
  • Mean time between calls: \text{E}(X) = \frac{1}{2} = 0.5 minutes
  • Variance of time between calls: \text{Var}(X) = \frac{1}{4} = 0.25 minutes^2

Visualization

A plot for an exponential distribution with \lambda = 1:

Here are the plots for the exponential distribution with \lambda = 1:

  • PDF (Probability Density Function): This shows the likelihood of different values of x. The curve decreases exponentially, indicating that smaller values of x (shorter waiting times) are more likely.
  • CDF (Cumulative Distribution Function): This represents the cumulative probability up to a certain value of x. The curve approaches 1 as x increases, indicating that the probability of the event occurring by time x increases over time.

See also the examples with probability calculations:

Examples of Exponential distribution

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