What is the expected value of an exponential distribution?

What is the expected value of an exponential distribution?

• A. 1/θ
• B. θ
• C. θ^2
• D. 2θ

Answer: (B)

The expected value of an exponential distribution is given by the parameter θ, where θ is the mean of the distribution. The exponential distribution is often used to model the time until an event occurs, such as the time until a radioactive particle decays or the time until a customer arrives at a service point. The probability density function (PDF) of the exponential distribution can be expressed as: \[ f(x; θ) = \frac{1}{θ} e^{-x/θ} \quad \text{for } x \geq 0 \] The expected value is derived from the integral of the product of \(x\) and the PDF over its entire range. Mathematically, the expected value \(E[X]\) can be computed as follows: \[ E[X] = \int_{0}^{\infty} x \cdot f(x) \, dx = \int_{0}^{\infty} x \cdot \left( \frac{1}{θ} e^{-x/θ} \right) \, dx \] Carrying out this integration leads to the result that the expected value is indeed \(θ\). This relationship underscores that in an exponential distribution, the mean value or the average outcome

The expected value of an exponential distribution is given by the parameter θ, where θ is the mean of the distribution. The exponential distribution is often used to model the time until an event occurs, such as the time until a radioactive particle decays or the time until a customer arrives at a service point.

The probability density function (PDF) of the exponential distribution can be expressed as:

[ f(x; θ) = \frac{1}{θ} e^{-x/θ} \quad \text{for } x \geq 0 ]

The expected value is derived from the integral of the product of (x) and the PDF over its entire range. Mathematically, the expected value (E[X]) can be computed as follows:

[ E[X] = \int_{0}^{\infty} x \cdot f(x) , dx = \int_{0}^{\infty} x \cdot \left( \frac{1}{θ} e^{-x/θ} \right) , dx ]

Carrying out this integration leads to the result that the expected value is indeed (θ).

This relationship underscores that in an exponential distribution, the mean value or the average outcome