Problem 31 - Calculus for Biology and Medicine

Question

Suppose that \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Show that \(E(X)=\mu\). [You may use the fact that if \(Z\) is standard normally distributed, then \(E(Z)=0\) and \(\operatorname{var}(X)=1 .]\)

Step-by-Step Solution

Answer

The expected value of a normally distributed variable \(X\) with mean \(\mu\) is \(E(X) = \mu\).

Step 1: Define the Normal Distribution

Given that the random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), we write \(X \sim N(\mu, \sigma^2)\). This means that any value of \(X\) follows the probability density function of a normal distribution with these parameters.

Step 2: Standardize the Variable

The random variable \(X\) can be standardized to \(Z\) using the transformation: \[Z = \frac{X - \mu}{\sigma}\]. This transformation aims to convert \(X\) into a standard normal variable, where \(Z\) is standard normally distributed with mean 0 and variance 1.

Step 3: Use Expectation Property of Standard Normal Variable

With \(Z\) as a standard normal variable, it is known that the expected value of \(Z\) is \(E(Z) = 0\). We now use this property to find \(E(X)\).

Step 4: Express Expectation of X in Terms of Z

By rearranging the standardized variable transformation, we have \[X = \sigma Z + \mu\]. Now, we express the expectation of \(X\) using this relationship: \[E(X) = E(\sigma Z + \mu)\].

Step 5: Apply Linearity of Expectation

Apply the linearity of expectation: \[E(X) = E(\sigma Z + \mu) = \sigma E(Z) + E(\mu)\]. Since \(E(\mu) = \mu\) (as it is a constant) and \(E(Z) = 0\), substitute these values to get: \[E(X) = \sigma \times 0 + \mu = \mu\].

Step 6: Conclusion

Thus, it is shown that the expected value, or mean, of a normally distributed random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\) is indeed \(E(X) = \mu\).

Key Concepts

Normal Distribution

The normal distribution, often known as the bell curve, is a fundamental concept in statistics and probability theory. It describes how the values of a random variable are distributed, assuming a symmetric, bell-shaped pattern.

When we say a random variable is normally distributed with parameters \( \mu \) and \( \sigma^2 \), we denote it as \( X \sim N(\mu, \sigma^2) \). Here, \( \mu \) represents the mean of the distribution, while \( \sigma^2 \) is the variance. The standard deviation, \( \sigma \), is the square root of the variance.

• Mean (\( \mu \)): Centrally locates the distribution.
• Variance (\( \sigma^2 \)): Describes the spread of the distribution, with larger variances indicating a more spread out distribution.

This distribution is crucial as many natural phenomena and statistical results follow it, allowing for straightforward analysis and prediction.

Standardization

Standardization is the process of converting a random variable into a standard normal variable. This is achieved by shifting and rescaling its values.

For a normally distributed variable \(X\) with a mean \(\mu\) and standard deviation \(\sigma\), the standardization transforms it into \(Z\), defined by the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
The result is a standard normal distribution with a mean of 0 and a variance of 1.

• This transformation simplifies problems, as it standardizes any normal distribution, making them easier to work with.
• It allows for comparison between different distributions by bringing them into a common framework.
• Helpful in probability calculations, since many statistical tables are based on the standard normal distribution.

Standardization is a key technique in data analysis, particularly when comparing scores from different normal distributions.

Linearity of Expectation

Linearity of expectation is a fundamental property of mathematical expectation. It states that the expected value of a sum of random variables is equal to the sum of their expected values.

For any random variables \( X \) and \( Y \), and constants \( a \) and \( b \), the following holds:
\[ E(aX + bY) = aE(X) + bE(Y) \]
This property is particularly useful in simplifying calculations involving random variables.

• Allows breaking down complex expectations into simpler parts.
• Useful in statistical and probabilistic analysis.

In the context of our problem, we apply linearity of expectation to find \(E(X)\) by writing \(X\) in terms of \(Z\) and \(\mu\), simplifying expectations using constants, and known expectations of the standard normal variable.

Mean of Random Variable

The mean of a random variable, also known as the expectation or expected value, provides a measure of central location for its distribution.

For a discrete random variable \(X\) with probability \(P(X=x_i)\), the mean is calculated as:
\[ E(X) = \sum x_i \cdot P(X = x_i) \]
For a continuous variable, it is calculated using integrals:
\[ E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx \]
The mean represents the "center of mass" or balancing point of the probability distribution.

• In the context of normal distribution, the mean \(\mu\) is both the average and the point of symmetry.
• By the linearity of expectation and properties of normally distributed variables, we find that \(E(X) = \mu\).

Understanding the mean of a random variable is essential for statistical inference, allowing us to make predictions about the variable within its distribution.