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 \( E(X) = \mu \) for a normally distributed variable with mean \( \mu \).

Step 1: Understanding the Problem

We need to show that if a random variable \( X \) is normally distributed with mean \( \mu \) and standard deviation \( \sigma \), then the expected value \( E(X) \) is equal to \( \mu \).

Step 2: Define the Transformation

Consider a standard normal variable \( Z \) that has mean 0 and standard deviation 1. We can express \( X \) as a transformation of \( Z \): \( X = \mu + \sigma Z \). This uses the properties of the normal distribution.

Step 3: Apply the Expectation Operator

Use the linearity of expectation to find \( E(X) \): \[ E(X) = E(\mu + \sigma Z) = E(\mu) + E(\sigma Z). \] Since \( \mu \) is a constant, \( E(\mu) = \mu \), and \( \sigma \) is a constant, \( E(\sigma Z) = \sigma E(Z) \).

Step 4: Substitute Known Values

We know that \( E(Z) = 0 \) by definition of the standard normal distribution. Substitute this into the equation from Step 3: \[ E(X) = \mu + \sigma \cdot 0 = \mu. \]

Step 5: Conclusion

We've shown that the expected value \( E(X) = \mu \) for a normally distributed variable \( X \) with mean \( \mu \) and standard deviation \( \sigma \).

Key Concepts

Normal Distribution

A normal distribution, often cited as the bell curve due to its distinctive shape, is a continuous probability distribution important in statistics. This curve is defined by two parameters:

• Mean (bc): The average or central point of the distribution.
• Standard Deviation (c3): Describes the spread or "width" of the distribution.

The curve is symmetrical around its mean, meaning values are equally spread out around the center, no matter the value of bc. Most values cluster around the mean, creating the bell shape, with fewer occurring as you move further away.
Due to its mathematical properties, normal distribution is widely used in statistical analysis, making it a staple in fields such as finance, research, and quality control. It serves as an underlying assumption for many statistical tests and models.

Linearity of Expectation

The linearity of expectation is a fundamental principle in probability theory, allowing us to easily calculate the expected value of a sum of random variables. This principle states that:

• The expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether they are independent or not.

Mathematically, if you have two random variables, say \( X \) and \( Y \), the linearity of expectation is expressed as:\[E(X + Y) = E(X) + E(Y)\]
This property greatly simplifies calculations, especially in scenarios where transformations are applied. For instance, in our problem, considering \( X = \mu + \sigma Z \), we use linearity of expectation to evaluate:\[E(X) = E(\mu + \sigma Z) = E(\mu) + E(\sigma Z)\]Since constants can be factored out of the expectation operator, the result simplifies to:\[E(X) = \mu + \sigma E(Z)\]Given that \( E(Z) = 0 \) (as \( Z \) is standard normal), illustrating that handling expectations is streamlined with this principle.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It's widely used as a reference in statistics when evaluating properties of normal variables due to its standardized form.
Since it's centered at zero, every point on the horizontal axis represents the number of standard deviations away from the mean. Thus, this makes it easier to compute probabilities and assess the behavior of data.
The transformation from a standard normal variable \( Z \) to a normal variable \( X \) is achieved through:\[X = \mu + \sigma Z\]This transformation reflects how standard normal variables scale and shift to fit any normal distribution. Thus, calculations like finding probabilities and expectations are performed in this standard setting and translated to any particular normal distribution by adjusting for \( \mu \) and \( \sigma \).
Understanding the standard normal distribution is essential for constructing confidence intervals, hypothesis testing, and normalization processes in datasets.