Exponential Distribution Calculator

What is the Exponential Distribution?

The exponential distribution is a continuous probability distribution that 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. It is often used to model the time until an event occurs, such as the time until a machine fails or the time between arrivals at a service center.

Formulas and Their Meanings

1. Probability Density Function (PDF): \[f(x) = \lambda e^{-\lambda x}, \quad x \geq 0\] Where \(\lambda\) is the rate parameter, and \(x\) is the random variable.

2. Cumulative Distribution Function (CDF): \[F(x) = 1 - e^{-\lambda x}, \quad x \geq 0\] This gives the probability that an event occurs within time \(x\).

3. Mean: \[E[X] = \frac{1}{\lambda}\] The expected value or average of the distribution.

4. Median: \[M = \frac{\ln(2)}{\lambda}\] The value that separates the higher half from the lower half of the distribution.

5. Variance: \[Var(X) = \frac{1}{\lambda^2}\] A measure of the spread of the distribution.

6. Standard Deviation: \[SD = \frac{1}{\lambda}\] The square root of the variance, giving a measure of spread in the same units as the original data.

Calculation Steps

1. Identify the rate parameter \(\lambda\) and the interval \([x_1, x_2]\).
2. Calculate the probability using the CDF: \(P(x_1 < X < x_2) = F(x_2) - F(x_1)\).
3. Compute the mean, median, variance, and standard deviation using the formulas above.

Example Calculation

Let's calculate for an exponential distribution with \(\lambda = 0.5\) and interval \([1, 3]\).

1. Probability: \(P(1 < X < 3) = (1 - e^{-0.5 \cdot 3}) - (1 - e^{-0.5 \cdot 1}) = 0.2325\)
2. Mean: \(E[X] = \frac{1}{0.5} = 2\)
3. Median: \(M = \frac{\ln(2)}{0.5} = 1.3863\)
4. Variance: \(Var(X) = \frac{1}{0.5^2} = 4\)
5. Standard Deviation: \(SD = \frac{1}{0.5} = 2\)

Visual Representation

This graph represents the probability density function of an exponential distribution. The shaded area illustrates the probability \(P(x_1 < X < x_2)\).