Problem 32 - Calculus for Biology and Medicine

Question

Suppose that \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Show that \(\operatorname{var}(X)=\sigma^{2} .\) [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

\(\operatorname{var}(X) = \sigma^2\) for normally distributed \(X\).

Step 1: Expressing the random variable X

We begin by expressing the normally distributed random variable \(X\). If \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), we can write it as \(X = \mu + \sigma Z\), where \(Z\) is a standard normal variable with mean 0 and variance 1.

Step 2: Calculate the Expected Value

Using the expression for \(X\), the expected value \(E(X)\) can be calculated as follows: \[E(X) = E(\mu + \sigma Z) = E(\mu) + E(\sigma Z).\] Since \(\mu\) is a constant, \(E(\mu) = \mu\). Also, \(\sigma\) is a constant, so \(E(\sigma Z) = \sigma E(Z) = \sigma \cdot 0 = 0\). Therefore, \[E(X) = \mu + 0 = \mu.\]

Step 3: Expression of Variance of X

The variance of \(X\) is \(\operatorname{var}(X) = \operatorname{var}(\mu + \sigma Z)\). Remember, variance properties dictate \(\operatorname{var}(aY + b) = a^2 \operatorname{var}(Y)\) for constant \(a\) and \(b\) and random variable \(Y\).

Step 4: Apply Variance Property

Applying the property of variance, we have: \[\operatorname{var}(\mu + \sigma Z) = \sigma^2 \operatorname{var}(Z) + \operatorname{var}(\mu).\] Since \(\mu\) is a constant, \(\operatorname{var}(\mu) = 0\). Thus, substituting the known variance \(\operatorname{var}(Z) = 1\), we get \[\operatorname{var}(X) = \sigma^2 \times 1 = \sigma^2.\]

Step 5: Conclusion

Thus, for a normally distributed variable \(X\) with standard deviation \(\sigma\), \(\operatorname{var}(X)\) is indeed \(\sigma^2\), corroborating the given properties of the normal distribution.

Key Concepts

Variance

Variance is a key concept in statistics, especially when dealing with normal distributions. It helps us understand how data points differ from the average or expected value. Essentially, variance measures the spread of a set of numbers. The more spread out these numbers are, the higher the variance.
To calculate variance, we use the formula: \[ \operatorname{var}(X) = E((X - \mu)^2) \]where:

• \(X\) represents our data or random variable
• \(\mu\) is the mean or expected value
• \(E\) denotes the expected value operator

In the case of a normal distribution, each random variable \(X\) is often expressed as a function of a standard normal variable \(Z\). By progressing through the calculations, we find that \(\operatorname{var}(X) = \sigma^2\), highlighting how crucial standard deviation \(\sigma\) is to determining variance. Since variance takes into account the differences squared, it always produces a non-negative value, indicating the extent of variability in the dataset.

Expected Value

The expected value is sometimes referred to as the mean or average of a random variable, providing a central point from which variance and standard deviation can be calculated. It signifies the center of a probability distribution and is critical for interpreting data in a meaningful way.
When \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), the expected value is simply \(\mu\). In mathematical terms:\[E(X) = \mu\]Using properties of expected values, if you have a constant \(a\) and a random variable \(Y\):

• \(E(a) = a\)
• \(E(aY) = aE(Y)\)

For a normal random variable \(X = \mu + \sigma Z\), where \(Z\) is standard normal with \(E(Z) = 0\), the expected value computation follows:\(E(X) = E(\mu + \sigma Z) = \mu + 0 = \mu\)This confirms that the expected value is indeed the mean of the normal distribution, making it a vital statistic for data analysis and prediction.

Standard Deviation

Standard deviation is a statistical measurement that describes the amount of variability or dispersion in a set of numbers. It is derived as the square root of variance, offering an intuitive measure of how spread out the numbers are around the mean. For a normal distribution, standard deviation is central to understanding how data is distributed.
In formulas, standard deviation \(\sigma\) is represented as:\[\sigma = \sqrt{\operatorname{var}(X)}\]where \(\operatorname{var}(X)\) is the variance. This relation indicates how standard deviation provides a scale for the data, allowing us to understand the spread in terms of original units.
The advantages of standard deviation include:

• It is unaffected by outliers as it focuses on overall data spread.
• It is used in numerous statistical analyses, including confidence intervals and hypothesis testing.

In normal distributions, the standard deviation determines the shape of the curve. A larger \(\sigma\) will result in a flatter, more spread-out distribution, while a smaller \(\sigma\) signifies a steeper, narrower distribution.