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 \(E(X)\) of an exponentially distributed random variable with parameter \(\lambda\) is \( \frac{1}{\lambda} \).

Step 1: Understand the Exponential Distribution

The exponential distribution with parameter \(\lambda\) has a probability density function given by \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\). It is used to model the time between events in a process that occurs continuously and independently at a constant average rate.

Step 2: Write the Formula for Expected Value

The expected value \(E(X)\) of a continuous random variable \(X\) with density function \(f(x)\) is defined as \( E(X) = \int_{0}^{\infty} x f(x) \, dx \). For an exponentially distributed random variable, substitute the probability density function \(f(x) = \lambda e^{-\lambda x}\) into this formula.

Step 3: Substitute and Simplify the Integral

Substitute the exponential probability density function into the expected value formula: \[E(X) = \int_{0}^{\infty} x \lambda e^{-\lambda x} \, dx\]Factor out \(\lambda\) since it is a constant: \[E(X) = \lambda \int_{0}^{\infty} x e^{-\lambda x} \, dx\]

Step 4: Use Integration by Parts

Apply integration by parts where \(u = x\) and \(dv = e^{-\lambda x} \, dx\). Then, \(du = dx\) and \(v = -\frac{1}{\lambda} e^{-\lambda x}\). The integration by parts formula is \(\int u \, dv = uv - \int v \, du\).

Step 5: Calculate the Integral

Substitute into the integration by parts formula:\[E(X) = \lambda \left[ -\frac{x}{\lambda} e^{-\lambda x} \bigg|_0^\infty + \int_{0}^{\infty} \frac{1}{\lambda} e^{-\lambda x} \, dx \right]\]Evaluate the boundary terms (they equal zero at both limits) and calculate the remaining integral:\[E(X) = \lambda \left[ 0 + \frac{1}{\lambda^2} \right] = \frac{1}{\lambda}\]

Step 6: Conclude the Calculation

Thus, the expected value of an exponentially distributed random variable with parameter \(\lambda\) is \(E(X) = \frac{1}{\lambda}\). This result indicates that the mean time between events is inversely proportional to the rate parameter.

Key Concepts

Exponential Distribution

The exponential distribution is a fundamental concept in probability that often models scenarios such as the time between arrivals of buses at a station or the time between transactions in a store. It's defined by a parameter \(\lambda\), which represents the rate at which the events occur. The higher the value of \(\lambda\), the more frequently the events happen. This distribution is continuous, meaning it can take on any positive real value.
The probability density function (PDF) for an exponential distribution is given by:

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

This formula shows that the probability of observing a particular value decreases exponentially as \(x\) increases. In other words, smaller intervals occur more frequently, which is expected when events happen independently and at a constant rate.

Probability Density Function

A probability density function (PDF) gives us a way to describe the likelihood of a continuous random variable assuming a particular value. Unlike discrete probabilities, which are calculated for specific outcomes, a PDF helps us understand the distribution of probabilities over a continuum of values. This is especially useful for modeling real-world phenomena that happen continuously.
For the exponential distribution specifically, the PDF is:

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

This function tells us how the probability is distributed over possible values of \(x\). Since a PDF is used with continuous variables, it doesn’t give a probability directly, but rather a density that needs to be integrated over an interval to give the probability of \(x\) falling within that range.

Integration by Parts

Integration by parts is a mathematical technique used to solve integrals where direct integration is not straightforward. This method is especially useful in cases where an integral involves a product of functions, such as \(x e^{-\lambda x}\). The rule is derived from the product rule for differentiation and can be expressed mathematically as: \[\int u \, dv = uv - \int v \, du\]To apply integration by parts, you select which part of the expression to differentiate (\(u\)) and which to integrate (\(dv\)). In the case of the exponential distribution's expected value calculation, \(u = x\) and \(dv = e^{-\lambda x} \, dx\) were chosen. This leads to \(du = dx\) and \(v = -\frac{1}{\lambda} e^{-\lambda x}\), allowing us to break down the integral into more manageable parts.

Random Variable

A random variable is a critical concept in probability theory and statistics that describes numerical outcomes of a random phenomenon. It is not a variable in the traditional algebraic sense; instead, it maps outcomes of statistical experiments to numerical values. Random variables can be discrete or continuous.
In the context of the exponential distribution, \(X\) represents the time between events in a Poisson process. Since it can take any non-negative real number, \(X\) is a continuous random variable. This characteristic allows it to be modeled using a probability density function, which can then be used to compute various probabilities and expected values, such as finding the mean time until the next event using the expected value.