Problem 33 - Calculus for Biology and Medicine

Question

Suppose that \(X\) is standard normally distributed. Find \(E(|X|)\).

Step-by-Step Solution

Answer

The expected value \(E(|X|)\) is \(\sqrt{\frac{2}{\pi}}\).

Step 1: Understand the Standard Normal Distribution

The standard normal distribution, denoted by the random variable \(X\), has a mean of 0 and a standard deviation of 1. It's represented by the probability density function (pdf) \(f(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\). We are tasked with finding the expected value of the absolute value of \(X\), noted as \(E(|X|)\).

Step 2: Define the Expected Value of |X|

The expected value of \(|X|\), where \(X\) has a standard normal distribution, can be expressed as \(E(|X|) = \int_{-\infty}^{\infty} |x| f(x) \, dx\). Here, we apply the pdf of the standard normal distribution.

Step 3: Break Down the Integral

Since the function \(|x|\) is even, the integral can be simplified as \(E(|X|) = 2\int_{0}^{\infty} x f(x) \, dx\). We multiply by 2 to account for both sides of the symmetric interval around the origin, as \(f(x)\) is also even.

Step 4: Substitute the PDF of Standard Normal

Substitute \(f(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\) into the integral: \(E(|X|) = 2 \int_{0}^{\infty} x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx\).

Step 5: Apply Standard Techniques

We can solve this integral using known results or transformation techniques. The integral \(\int_{0}^{\infty} x e^{-\frac{x^2}{2}} \, dx\) can be solved using substitution or referencing standard results for Gaussian integrals. This is a known integral and its solution is \(\sqrt{\frac{\pi}{2}}\).

Step 6: Compute the Final Result

Using the result from the previous step: \(E(|X|) = \frac{2}{\sqrt{2\pi}} \times \sqrt{\frac{\pi}{2}} = \sqrt{\frac{2}{\pi}} \times \frac{\sqrt{\pi}}{\sqrt{2}} = \sqrt{\frac{\pi}{2}}\). Therefore, the expected value is indeed \(\sqrt{\frac{2}{\pi}}\).

Key Concepts

Expected Value

The expected value, often denoted as \(E(X)\), is a fundamental concept in probability and statistics, describing the average or mean value of a random variable over numerous instances. For the standard normal distribution, \(X\) is symmetrically distributed around its mean of 0. In our specific problem, we're interested in the expected value of the absolute value of \(X\), represented as \(E(|X|)\). This is crucial because it accounts for the magnitude of \(X\) regardless of direction (positive or negative).

To compute \(E(|X|)\), we integrate the function \(|x| f(x)\), where \(f(x)\) is the probability density function of the standard normal distribution. The formula becomes:

• \(E(|X|) = \int_{-\infty}^{\infty} |x| f(x) \, dx\)

The symmetry of the distribution simplifies our task, allowing us to focus only on the positive side of the axis by multiplying by 2, hence:

• \(E(|X|) = 2\int_{0}^{\infty} x f(x) \, dx\)

Ultimately, finding this expected value provides a sense of the average absolute deviation of \(X\) from the mean in a standard normal framework.

Probability Density Function

A probability density function (PDF) defines how the probability of a continuous random variable is distributed over various values. For the standard normal distribution, the PDF is particularly important since it's one of the most used distributions due to its properties.

The formula for the probability density function of a standard normal distribution is as follows:

• \(f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\)

This function describes a bell-shaped curve that perfectly reflects the symmetry and spread around the mean (0 in this case), with a standard deviation of 1.

When calculating an integral related to the expected value under this distribution, such as \(E(|X|)\), this PDF is essential. It reflects how probability decreases as \(x\) moves away from the mean. By integrating this PDF across the range of interest, we can determine various characteristics of the distribution, such as mean or expected value.
Understanding the PDF allows us to realize how probabilities are assigned to continuous outcomes, guiding both theoretical analyses and practical applications.

Gaussian Integral

The concept of the Gaussian integral surfaces frequently in probability and calculus, especially concerning normal distributions. In the context of our exercise, the Gaussian integral helps evaluate integrals involving the standard normal PDF.

A Gaussian integral is typically characterized by the expression:

• \( \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} \)

When multiplied by variables or constants, different forms of these integrals arise, and they often evaluate to known results, essential in solving practical mathematical problems.

In our task of finding \(E(|X|)\), we simplify to:

• \(2 \int_{0}^{\infty} x e^{-\frac{x^2}{2}} \, dx \)

This integral relates directly to known Gaussian integral results, allowing us to efficiently find our expected value using the solved integral part, yielding \(\sqrt{\frac{\pi}{2}}\) as an intermediate step.
Efficient computation of Gaussian integrals thus plays a crucial role in evaluating more complex probability-related calculations.