Problem 121 - An Introduction to Mathematical Statistics and Its Applications

Question

If \(E(W)=\mu\) and \(\operatorname{Var}(W)=\sigma^{2}\), show that \(E\left(\frac{W-\mu}{\sigma}\right)=0\) and \(\operatorname{Var}\left(\frac{W-\mu}{\sigma}\right)=1\)

Step-by-Step Solution

Answer

After applying the properties of expectation and variance, it was found that \(E\left(\frac{W-\mu}{\sigma}\right)=0\) and \(Var\left(\frac{W-\mu}{\sigma}\right)=1\)

Step 1: Establish Expected Value

Firstly, apply the linearity of expectation law to \(E\left(\frac{W-\mu}{\sigma}\right)\). This means you can break down the expected value operation in two: \(E\left(\frac{W}{\sigma}\right) - E\left(\frac{\mu}{\sigma}\right)\). This formula simplifies to \(\frac{E(W)}{\sigma} - \frac{\mu}{\sigma}\). Knowing that \(E(W)=\mu\), we substitute and obtain \(\frac{\mu}{\sigma} - \frac{\mu}{\sigma}\) which equals 0.

Step 2: Find Variance

Then calculate the variance by applying the formula \(Var(aX) = a^{2}Var(X)\) where \(a= \frac{1}{\sigma}\) and \(X = W-\mu\). This results to \(Var\left(\frac{W-\mu}{\sigma}\right)=\left(\frac{1}{\sigma}\right)^{2} Var(W-\mu)\). Because for any random variable \(X\) and constant \(c\), \(Var(X+c) = Var(X)\), we can say \(Var(W-\mu) = Var(W)\) . Therefore, substituting \(\sigma^{2}\) for \(Var(W)\), we thus have \(\left(\frac{1}{\sigma}\right)^{2} \cdot \sigma^{2}=1\)

Step 3: Conclusion

In conclusion, after application of expectation and variance properties, it has been shown that \(E\left(\frac{W-\mu}{\sigma}\right)=0\) and \(Var\left(\frac{W-\mu}{\sigma}\right)=1\)

Key Concepts

Expected Value

The expected value, also known as the mean, is a fundamental concept in mathematical statistics that sums up the center or average of a probability distribution. To understand it, visualize rolling a dice: each number has an equal chance of coming up. The expected value would be the average of all these possible outcomes, calculated by multiplying each outcome by its probability and summing them up.

In our exercise, the expected value, denoted as \( E(W) \), is the average value we would expect for our random variable \( W \) if we were to observe it infinitely many times. In the case of standardization, we are interested in transforming \( W \) into a new variable \( Z \)=\( \frac{W - \mu}{\sigma} \) such that its expected value is zero. This transformation is akin to shifting the original distribution so that its mean is at zero, which simplifies further analyses, especially in comparing different random variables.

Variance

Variance is a measure of spread or dispersion in a set of data points, specifically, how much the values differ from the mean. In simpler terms, high variance means that the data points are scattered widely from the average, and low variance indicates the opposite.

Mathematically, variance is defined as the average of the squared differences from the mean, which can be calculated for a random variable \( W \) as \( \sigma^2 = \operatorname{Var}(W) \). In the context of our exercise, understanding variance is crucial because it helps us quantify the volatility or risk associated with the random variable. When we standardize the random variable \( W \) to create \( Z \), we are essentially scaling it by its standard deviation \( \sigma \) to achieve a variance of 1. This means that \( Z \) will have the same shape as the distribution of \( W \) but will be scaled to have a uniform dispersion, which is beneficial in statistical analysis and comparisons.

Standardization of Random Variables

Standardization is a method to convert a random variable into a standard form that has an expected value of 0 and a variance of 1. This technique is especially useful when comparing different variables that are not initially on the same scale or when we need to apply certain statistical methods that assume data with these standardized properties.

The step-by-step solution shows the process of standardizing a random variable \( W \). By subtracting the mean, \( \mu \) from \( W \) and dividing by the standard deviation, \( \sigma \) we are not only centering the data around 0 but also scaling it so that its units are expressed in terms of standard deviation. This transformation produces a new random variable, usually denoted as \( Z \) that is dimensionless and has properties that make it universally comparable to other standardized variables. It's essential for various applications like hypothesis testing and creating z-scores, which measure how far away a single data point is from the mean in terms of standard deviations.