Problem 80 - An Introduction to Mathematical Statistics and Its Applications

Question

Show that the expected value associated with the exponential distribution, \(f_{Y}(y)=\lambda e^{-\lambda y}, y>0\), is \(1 / \lambda\), where \(\lambda\) is a positive constant.

Step-by-Step Solution

Answer

The expected value of the exponential distribution \(f_{Y}(y)=\lambda e^{-\lambda y}, y>0\), is \(1 / \lambda\). It is shown via the computation of the integral of y times the pdf using integration by parts.

Step 1: Expected Value Formula

The expected value of a continuous random variable Y, denoted as E(Y), is given by the formula: \(E(Y) = \int_{-\infty}^{+\infty} y f_Y(y) dy \). For an exponential distribution which only exists for \(y > 0\), this simplifies to \(E(Y) = \int_{0}^{+\infty} y \lambda e^{-\lambda y} dy \).

Step 2: Integration by Parts

To solve the integral, apply the method of integration by parts. Invoke the formula: \(\int u dv = uv - \int v du \). Let \(u=y\) and \(dv=\lambda e^{-\lambda y} dy\). Apply these to the integral formula.

Step 3: Evaluate Integral

After handling integration by parts, compute the anti-derivatives and calculating the integral between 0 and +∞. Finally, this calculation will yield \(1 / \lambda\).

Key Concepts

Integration by Parts

To grasp the concept of integration by parts, think of it as the reverse process of the product rule used in differentiation. It's a method usually employed when you are faced with the product of two functions that cannot be easily integrated as they are. The basic formula is \[ \int u dv = uv - \int v du \.\] You choose one part of your function to be \(u\) and the other to be \(dv\), and then differentiate and integrate accordingly to fill the parts of the formula. A good rule of thumb for choosing \(u\) in an exponential distribution problem like the one we have, is to pick the part that, when differentiated, simplifies the integral. This technique often turns a challenging integral into a simpler one that can be solved with basic integration methods.

Continuous Random Variable

A continuous random variable, unlike a discrete random variable, can take on an infinite number of values. For example, it can represent any value within a range of real numbers. The exponential distribution is a prime example of a continuous distribution, as it describes time until an event occurs, and time can take on any positive real value. For a continuous random variable, we use a probability density function (pdf) to define the likelihood of the variable falling within a particular range. When working with the exponential distribution \( f_Y(y) = \[\lambda e^{-\lambda y}, y > 0\]\), the \(\lambda\) parameter shapes the distribution and is representative of the rate at which events occur.

Expected Value Formula

The expected value of a continuous random variable provides a measure of the 'center' of the distribution, which you can loosely think of as a long-term average. The formula for the expected value of a continuous random variable is an integral that multiplies each outcome by its probability and sums these products over the entire range of possible outcomes. Expressed mathematically, it's \[ E(Y) = \int_{-\infty}^{+\infty} y f_Y(y) dy\]. In the case of the exponential distribution, the expected value integral simplifies, as the function is only non-zero for positive values of \(y\). Hence, the formula for our exercise contracts to \[E(Y) = \int_{0}^{+\infty} y \lambda e^{-\lambda y} dy\]. Through integration by parts, we can resolve this integral to find the expected value, as has been done in the exercise.