Question

Find the expected value and variance of a random variable, $$ \mathrm{Y}=\mathrm{a}_{1} \mathrm{X}_{1}+\mathrm{a}_{2} \mathrm{X}_{2}+\ldots \ldots+\mathrm{a}_{\mathrm{n}} \mathrm{X}_{\mathrm{n}} $$ where the \(\mathrm{X}_{\mathrm{i}}\) are independent and each have mean \(\mu\) and variance \(\sigma^{2}\). The \(a_{i}\) are constants.

Short answer

The expected value and variance of the random variable Y are: Expected value: \(E[Y] = \mu(a_1 + a_2 + \cdots + a_n)\) Variance: \(Var(Y) = \sigma^2(a_1^2 + a_2^2 + \cdots + a_n^2)\)

Step-by-step solution

Write the random variable Y as a sum of its components

Define the random variable Y as the linear combination: \(Y = a_1X_1 + a_2X_2 + \cdots + a_nX_n\)

Find the expected value of Y

Use the linearity property of expectation, which states that \(E[aX + bY] = aE[X] + bE[Y]\) for any constants a and b and random variables X and Y. Apply this property to the given random variable Y: \(E[Y] = E[a_1X_1 + a_2X_2 + \cdots + a_nX_n]\) \(E[Y] = a_1E[X_1] + a_2E[X_2] + \cdots + a_nE[X_n]\) Since the mean of each random variable \(X_i\) is \(\mu\), we can rewrite the equation as: \(E[Y] = a_1\mu + a_2\mu + \cdots + a_n\mu = \mu(a_1 + a_2 + \cdots + a_n)\)

Find the variance of Y

Use the property of variance for independent random variables, which states that \(Var(aX + bY) = a^2Var(X) + b^2Var(Y)\) for any constants a and b and independent random variables X and Y. Apply this property to the given random variable Y: \(Var(Y) = Var(a_1X_1 + a_2X_2 + \cdots + a_nX_n)\) \(Var(Y) = a_1^2Var(X_1) + a_2^2Var(X_2) + \cdots + a_n^2Var(X_n)\) Since the variance of each random variable \(X_i\) is \(\sigma^2\), we can rewrite the equation as: \(Var(Y) = a_1^2\sigma^2 + a_2^2\sigma^2 + \cdots + a_n^2\sigma^2 = \sigma^2(a_1^2 + a_2^2 + \cdots + a_n^2)\)

Present the results

We have calculated the expected value and variance of the random variable Y as: Expected value: \(E[Y] = \mu(a_1 + a_2 + \cdots + a_n)\) Variance: \(Var(Y) = \sigma^2(a_1^2 + a_2^2 + \cdots + a_n^2)\)

Key concepts

Random Variables

In probability and statistics, a random variable represents a quantitative feature whose value depends on the outcomes of a random phenomenon. For instance, tossing a coin results in two possible outcomes: heads or tails. If we assign a value such as 1 to heads and 0 to tails, we have created a random variable that interprets the outcome of the toss.

A random variable can be discrete, taking on specific isolated values like the toss of a die, or continuous, such as the temperature on a given day. Understanding random variables is crucial as they are foundational in defining and calculating statistics like means (or expected values), variances, and probabilities of different outcomes.

Linearity Property of Expectation

The linearity property of expectation is a fundamental concept in statistics that simplifies the calculation of the expected value for a combination of random variables. According to this property, the expected value of a sum (or linear combination) of random variables is equal to the sum of their individual expected values, regardless of whether they are independent or not.

Mathematically, for constants \(a\) and \(b\) and random variables \(X\) and \(Y\), this is expressed as:\
\[ E[aX + bY] = aE[X] + bE[Y] \]
This property is powerful because it allows us to perform calculations on the expected values of complex expressions by breaking them down into simpler parts.

Variance Properties

Variance measures how much a set of random variables deviates from their expected value, indicating the spread of a distribution. A key property of variance is that, unlike expectation, it is not a linear operation, but it does hold a special property for independent random variables.

Specifically, for any constants \(a\) and \(b\), and for independent random variables \(X\) and \(Y\), the variance of their linear combination can be expressed as:\
\[ Var(aX + bY) = a^2Var(X) + b^2Var(Y) \]
For independent variables, variances add up after being scaled by the square of the constants used in their linear combination, signifying that the overall variability increases with the square of the scaling factors. This property is extremely useful when handling variables that are affected by different scales of influence.

Independent Random Variables

Random variables are independent if the occurrence of one does not affect the probability distribution of the others. This concept is essential when dealing with multiple random processes, as independence simplifies calculations and allows the use of the variance property mentioned earlier.

Independence is a strong assumption that may not hold in all cases, especially in complex systems where variables often interact. However, when confirmed, it enables us to treat individual random variables in a vacuum for various predictive and analytical calculations, making the process more manageable.

Linear Combinations of Random Variables

A linear combination of random variables occurs when these variables are summed together, each being multiplied by a constant. This is a common operation when constructing new variables from existing ones, such as creating an index or score.

The importance of a linear combination lies in its ability to integrate distinct random processes into a single measure, which can then be analyzed using the expectations and variances of the original variables. In the context of the exercise mentioned, understanding that the expected value of a linear combination is the weighted sum of the expected values, and the variance is the sum of the squared weights times the variances, provided the variables are independent, greatly simplifies the process of assessing the behavior of complex models.