Chapter 15: Problem 13

Show that adding a constant \(K\) to a random variable increases the average by \(K\) but does not change the variance. Show that multiplying a random variable by \(K\) multiplies both the average and the standard deviation by \(K\).

Short Answer

Expert verified

Adding \(K\) increases the mean by \(K\) but keeps the variance the same. Multiplying by \(K\) scales both mean and standard deviation by \(K\).

Step by step solution

Define the random variable and its properties

Let the random variable be denoted as \(X\) with the mean (expected value) \(E(X) = \mu_X\) and variance \(Var(X) = \sigma_X^2\). We'll consider two operations: adding a constant \(K\) and multiplying by a constant \(K\).

Adding a constant to the random variable

Consider the new random variable \(Y = X + K\). The mean of \(Y\) can be calculated as follows: \[E(Y) = E(X + K) = E(X) + K = \mu_X + K\]. Therefore, the mean of \(Y\) increases by \(K\).

Verify the variance when adding a constant

The variance of \(Y\) is calculated by: \[Var(Y) = Var(X + K) = Var(X) = \sigma_X^2\]. Constant \(K\) does not affect the variance.

Multiplying the random variable by a constant

Next, consider \(Z = KX\). The mean of \(Z\) is: \[E(Z) = E(KX) = K \times E(X) = K \times \mu_X\]. Hence, multiplying \(X\) by \(K\) multiplies its mean by \(K\).

Verify the variance when multiplying by a constant

The variance of \(Z\) is: \[Var(Z) = Var(KX) = K^2 \times Var(X) = K^2 \times \sigma_X^2\]. The standard deviation of \(Z\) is \sqrt{Var(Z)} = K \times \sigma_X\. Hence, multiplying \(X\) by \(K\) multiplies its standard deviation by \(K\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mean of random variables

When dealing with random variables, understanding their mean (or expected value) is essential. The mean of a random variable provides a measure of its central tendency. It is similar to the average in a set of numbers and helps to identify the 'center' or 'typical' value around which the random variable tends to cluster. For a random variable \(X\), the mean is denoted as \(E(X)\) or \(\mu_X\).

Let's break it down:

- **Adding a Constant**: When you add a constant \(K\) to a random variable, the entire distribution shifts by that constant value. If \(Y = X + K\), then the mean of \(Y\) can be calculated as:

\[ E(Y) = E(X + K) = E(X) + K = \mu_X + K \]

This shows that the mean increases by the constant \(K\), but the shape of the distribution doesn't change.

- **Multiplying by a Constant**: When you multiply a random variable by a constant, the mean is scaled by that same constant. If \(Z = KX\), the mean of \(Z\) would be:

\[ E(Z) = E(KX) = K \times E(X) = K \times \mu_X \]

Thus, the mean is multiplied by \(K\). This operation scales all values of the random variable, but the 'center' of the distribution is adjusted accordingly.

variance of random variables

The variance of a random variable measures how spread out the values of the variable are around the mean. It gives us an idea about the variability or dispersion of the data. For a random variable \(X\), the variance is denoted as \(Var(X)\) or \(\sigma_X^2\).

Let's dive in:

- **Adding a Constant**: When a constant \(K\) is added to a random variable, the spread of the data does not change, only its position does. For \(Y = X + K\), the variance remains the same:

\[ Var(Y) = Var(X + K) = Var(X) = \sigma_X^2 \]

Thus, adding a constant does not change the variance.

- **Multiplying by a Constant**: When a random variable is multiplied by a constant, the variance is affected by the square of that constant. For \(Z = KX\), the variance of \(Z\) is given by:

\[ Var(Z) = Var(KX) = K^2 \times Var(X) = K^2 \times \sigma_X^2 \]

This means the variance is scaled by \(K^2\), indicating a larger spread if \(K > 1\) or a smaller spread if \(K < 1\).

standard deviation of random variables

The standard deviation of a random variable is the square root of its variance and offers a way of understanding the spread of the distribution in the same units as the random variable itself. For a random variable \(X\), the standard deviation is denoted as \(\sigma_X\).

Here's the breakdown:

- **Adding a Constant**: As with variance, adding a constant to a random variable does not change its standard deviation. For \(Y = X + K\):

\[ \sigma_Y = \sigma_X \]

The spread of the data remains the same; only the position of the data set shifts.

- **Multiplying by a Constant**: When multiplying a random variable by a constant, the standard deviation is scaled directly by that constant. For \(Z = KX\):

\[ \sigma_Z = K \times \sigma_X \]

Thus, the standard deviation is multiplied by \(K\), showing a direct scaling of the spread of the data based on the constant multiplier. This is particularly useful in fields like finance, where scaling risks and returns are common.