Further Mathematics | Combinations of random variables

Combinations of Random Variables: Mastering the Uncertainty

Hello there! Welcome to a crucial chapter in Further Mathematics Statistics. This topic, Combinations of Random Variables, might sound complicated, but it's fundamentally about managing uncertainty when events are linked.

Think of it this way: If you buy two different items, each with an uncertain price (a random variable), what is the expected total cost, and how spread out (variable) is that total cost likely to be? We need mathematical rules to combine these uncertainties accurately.

Don't worry if this seems tricky at first. We will break down every rule into simple, manageable steps. By the end of these notes, you'll be a pro at handling combined means and variances!

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1. The Expected Value (Mean): The Simple Rules

The Expected Value, \(E(X)\), is simply the theoretical long-run average, or the mean, of a random variable \(X\). When combining variables, calculating the expected value is usually very straightforward because of a concept called Linearity.

The great news is that the rules for Expected Values work whether the variables are independent or dependent!

Rule 1: Multiplying by a Constant (\(a\))

If you multiply a random variable \(X\) by a constant \(a\), the average value is multiplied by \(a\).

• The Rule: \(E(aX) = a E(X)\)

Example: If the average cost of a cinema ticket (\(X\)) is £8, then the average cost of buying 5 tickets is \(E(5X) = 5 E(X) = 5 \times 8 = £40\).

Rule 2: Adding a Constant (\(c\))

If you add a fixed amount \(c\) to a random variable \(X\), the average value increases by \(c\).

• The Rule: \(E(X + c) = E(X) + c\)

Example: If the average journey time (\(X\)) is 40 minutes, and you always spend 5 minutes finding parking (\(c\)), the average total time is \(E(X+5) = E(X) + 5 = 40 + 5 = 45\) minutes.

Rule 3: Combining Two Variables (X and Y)

To find the expected value of a combination of two variables, simply combine their individual expected values, keeping the constants and signs the same.

• The Rule: \(E(aX \pm bY) = a E(X) \pm b E(Y)\)

This rule applies whether you are adding (\(+\)) or subtracting (\(-\)) the variables.

Key Takeaway for Expected Values: Expected values are simple and linear. You can pull the constants out and keep the signs the same. \(E(\text{sum}) = \text{sum of } E\).

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2. The Variance Rules: Handling the Spread

This is where the rules become a little different, and it's essential to pay close attention! The Variance, \(Var(X)\), measures the spread or variability of a random variable.

!!! Independence Warning !!!
The rules for combining variances that we use in this syllabus only work if the random variables \(X\) and \(Y\) are independent. If they are not independent, the maths becomes much more complex (involving covariance), which is usually outside the scope of this unit. In your exams, you can assume independence unless stated otherwise.

Rule 1: Multiplying by a Constant (\(a\))

If you multiply \(X\) by a constant \(a\), the variance changes by \(a^2\). Why squared? Because variance is measured in squared units.

• The Rule: \(Var(aX) = a^2 Var(X)\)

Analogy: Imagine a piece of string of random length \(X\). If you double its length (\(2X\)), the expected length doubles, but the potential spread (variance) doesn't just double; it becomes four times as large, because the variation is amplified in two dimensions (squared).

Rule 2: Adding a Constant (\(c\))

Adding a fixed number \(c\) to every value does not change the spread of the data.

• The Rule: \(Var(X + c) = Var(X)\)

Example: If the costs of a meal (\(X\)) are spread between £10 and £20 (variance \(V\)), and you add a fixed £5 tip (\(X+5\)), the costs are now spread between £15 and £25. The centre shifts, but the width of the spread remains exactly the same (\(V\)).

Rule 3: Combining Two Variables (X and Y)

This is the single most important rule in this chapter: When combining independent variables, variances always ADD.

Why? Uncertainty accumulates. Whether you are summing two random values or finding the difference between them, the total uncertainty in the result is always greater than the uncertainty in either part alone.

• Addition Rule: \(Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)\)
• Subtraction Rule: \(Var(aX - bY) = a^2 Var(X) + b^2 Var(Y)\)

Notice that the formula for subtraction is exactly the same as the formula for addition!

Example: If the uncertainty (variance) in delivery time \(X\) is 4 minutes squared, and the uncertainty in payment time \(Y\) is 9 minutes squared, the total uncertainty in the process \(X+Y\) is \(4+9=13\). If you were looking at the difference in those times \(X-Y\), the uncertainty is still \(4+9=13\)!

Key Takeaway for Variance: Variance is always positive (because of the squaring) and always added (because uncertainty accumulates). Remember the \(a^2\) factor!

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3. Standard Deviation (\(\sigma\))

The Standard Deviation (\(\sigma\)) is simply the square root of the variance. It is usually easier to interpret because it is measured in the same units as the random variable itself.

The Calculation Process

You must calculate the combined variance first before finding the standard deviation.

1. Use the rules to find the combined variance: \(Var(aX \pm bY)\).
2. Take the positive square root of the result: \(\sigma_{(aX \pm bY)} = \sqrt{Var(aX \pm bY)}\).

You cannot just add or subtract standard deviations!

Did you know? Statisticians use variance for mathematical convenience (because the rules are simpler with the squares), but they convert back to standard deviation for real-world reporting so that the numbers make sense to non-statisticians.

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4. Generalising the Rules and Common Traps

Handling Multiple Variables

The rules extend easily to more than two variables (e.g., \(X_1, X_2, X_3, \dots, X_n\)).

• \(E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3)\)
• \(Var(X_1 + X_2 + X_3) = Var(X_1) + Var(X_2) + Var(X_3)\) (Assuming independence)

Crucial Distinction: \(n\) times a variable vs. Summing \(n\) independent copies

This is the most frequent source of error in exam questions! You must know the difference between \(nX\) and the sum of \(n\) independent random variables, \(X_1 + X_2 + \dots + X_n\).

Suppose \(X_1, X_2, X_3\) are three independent observations of the variable \(X\), where \(E(X) = \mu\) and \(Var(X) = \sigma^2\).

Case A: \(3X\) (3 times the same variable)
This represents a single value tripled (e.g., tripling the dimensions of one box).

• \(E(3X) = 3 E(X) = 3\mu\)
• \(Var(3X) = 3^2 Var(X) = 9\sigma^2\)

Case B: \(X_1 + X_2 + X_3\) (Sum of 3 independent variables)
This represents the total of three separate, independent values (e.g., the total weight of three independently chosen boxes).

• \(E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3) = \mu + \mu + \mu = 3\mu\)
• \(Var(X_1 + X_2 + X_3) = Var(X_1) + Var(X_2) + Var(X_3) = \sigma^2 + \sigma^2 + \sigma^2 = 3\sigma^2\)

Summary of the Distinction:

• When you multiply one variable by 3, the variance is multiplied by \(3^2 = 9\).
• When you sum 3 independent copies of a variable, the variance is multiplied by 3.

5. Worked Example & Final Summary

Example Scenario

A company manufactures components. The time (in minutes) taken to produce Component A (\(A\)) and Component B (\(B\)) are independent random variables.

• \(E(A) = 15\), \(Var(A) = 4\)
• \(E(B) = 20\), \(Var(B) = 9\)

A process requires manufacturing three A components and one B component. The total time \(T\) is \(T = 3A + B\). Find \(E(T)\) and \(Var(T)\).

Step-by-Step Solution

Part 1: Expected Time \(E(T)\)

1. Apply the linearity rule: \(E(3A + B) = E(3A) + E(B)\)
2. Apply Rule 1 (constant multiple): \(E(3A) = 3 E(A)\)
3. Calculation: \(E(T) = 3(15) + 20 = 45 + 20 = 65\) minutes.

Part 2: Variance of Time \(Var(T)\)

1. Apply the variance addition rule (crucially, we always add): \(Var(3A + B) = Var(3A) + Var(B)\)
2. Apply Rule 1 (squaring the constant): \(Var(3A) = 3^2 Var(A)\)
3. Calculation: \(Var(T) = (3^2 \times 4) + 9 = (9 \times 4) + 9 = 36 + 9 = 45\) minutes squared.

If the question asked for the standard deviation, you would calculate: \(\sigma(T) = \sqrt{45} \approx 6.71\) minutes.

Congratulations! You've covered the core algebraic rules for combining random variables. Practice these four rules diligently, especially the variance rules, and you will master this section of Statistics 3!