Problem 47 - Calculus for Biology and Medicine

Question

Let \(X\) be exponentially distributed with parameter \(\lambda\). Find \(E(X)\)

Step-by-Step Solution

Answer

The expected value of \(X\) is \(\frac{1}{\lambda}\).

Step 1: Understanding the Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events. When a random variable \(X\) is exponentially distributed, it is characterized by the parameter \(\lambda > 0\), which is the rate of the distribution.

Step 2: Recalling the Expected Value Formula for Exponential Distribution

For a random variable \(X\) that is exponentially distributed with parameter \(\lambda\), the expected value \(E(X)\) is given by the formula \(E(X) = \frac{1}{\lambda}\). This is a fundamental property of the exponential distribution, based on its probability density function and moment generating function.

Step 3: Substituting the Parameter into the Expected Value Formula

Since \(X\) is distributed with parameter \(\lambda\), we use the formula \(E(X) = \frac{1}{\lambda}\) and substitute \(\lambda\) as specified by the distribution of \(X\).

Step 4: Conclusion

The expected value of an exponentially distributed random variable \(X\) with parameter \(\lambda\) is \(\frac{1}{\lambda}\). Therefore, \(E(X) = \frac{1}{\lambda}\).

Key Concepts

Expected Value

The expected value of a random variable, often symbolized as \(E(X)\), is a key concept in probability and statistics. It represents the long-term average or mean of a random variable over numerous trials or experiments. For the exponential distribution, the expected value is directly linked to its parameter, \(\lambda\). The exponential distribution is unique because, unlike other distributions, it models the time until the next event occurs, such as the time between arrivals at a service center.

For an exponential distribution with parameter \(\lambda\), the formula for the expected value is relatively straightforward:

• \(E(X) = \frac{1}{\lambda}\)

This formula arises from the exponential distribution's probability density function. The expected value formula \(\frac{1}{\lambda}\) highlights the relationship between the average waiting time and the rate \(\lambda\). If the rate \(\lambda\) is high, events occur frequently, and thus, the expected time \(E(X)\) is short.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes. For continuous random variables, like those following an exponential distribution, this is often represented by a probability density function (PDF).

The exponential probability distribution is defined by its PDF:

• \(f(x; \lambda) = \lambda e^{-\lambda x}\) for \(x \geq 0\)

This function shows a decreasing pattern, reflecting that it's less likely for events to take a long time to occur than a short one.

Key properties of the exponential distribution include:

• Memoryless property: The probability of an event occurring in the future is independent of any past events.
• Non-negative values: It only takes values from \(0\) to infinity.

By knowing the parameter \(\lambda\), the distribution can model various real-world scenarios efficiently.

Moment Generating Function

The moment-generating function (MGF) is a powerful tool in probability theory. It provides a convenient way to derive the moments of a random variable, such as the mean and variance.

For an exponentially distributed random variable with rate \(\lambda\), the MGF is given by:

• \(M_X(t) = \frac{\lambda}{\lambda - t}\) for \(t < \lambda\)

Using the MGF, we can derive the expected value and variance.

The first moment, or the expected value \(E(X)\), can be found by taking the first derivative of the MGF with respect to \(t\) and evaluating at \(t = 0\). This process reaffirms the expected value \(\frac{1}{\lambda}\).

The moment-generating function is especially beneficial because it simplifies working with multiple and independent random variables, helping to combine them into one distribution efficiently. For exponential distributions, it highlights its main properties and aids in underpinning the mathematical applications.