Linear Combinations of Random Variables (College Board AP® Statistics)

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Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Linear combinations of random variables

What is a linear combination of random variables?

A linear combination of random variables is where multiples of given random variables are added together
If $X$ and $Y$ are random variables, then a linear combination is of the form $W = a X + b Y$
- where $a$ and $b$ are real constants
Linear combinations can model real-life scenarios
- e.g. on a given day, $X$ laptops are sold at $400 each and $Y$ desktops are sold at $600 each
  - the total revenue for a given day, $R$ , can therefore be modeled as $R = 400 X + 600 Y$

How do I find the mean of a linear combination of random variables?

To find the mean of a linear combination of random variables
- treat the original means just like the values of the variables
  - multiply each mean by the relevant constant
  - then add together
If $W = a X + b Y$ then $μ_{W} = a μ_{X} + b μ_{Y}$
If $W = a X - b Y$ then $μ_{W} = a μ_{X} - b μ_{Y}$
This can be extended to multiple random variables
- If $Y = a_{1} X_{1} \pm a_{2} X_{2} \pm . . . \pm a_{n} X_{n}$ then $μ_{Y} = a_{1} μ_{1} \pm a_{2} μ_{2} \pm . . . \pm a_{n} μ_{n}$

How do I find the variance of a linear combination of random variables?

To find the variance of a linear combination of independent random variables
- multiply each of the original variances by the relevant constant squared
- then add together
If $W = a X + b Y$ then ${σ_{W}}^{2} = a^{2} {σ_{X}}^{2} + b^{2} {σ_{Y}}^{2}$
If $W = a X - b Y$ then ${σ_{W}}^{2} = a^{2} {σ_{X}}^{2} + b^{2} {σ_{Y}}^{2}$
- The terms are always added within the formula
  - This is because the square of any negative number is always positive
This can be extended to multiple independent random variables
- If $Y = a_{1} X_{1} \pm a_{2} X_{2} \pm . . . \pm a_{n} X_{n}$ then ${σ_{Y}}^{2} = {a_{1}}^{2} {σ_{1}}^{2} + {a_{2}}^{2} {σ_{2}}^{2} + . . . + {a_{n}}^{2} {σ_{n}}^{2}$

Examiner Tips and Tricks

The formula for the variance of a linear combination only works if the random variables are independent. If you use this formula in an exam question, be sure to state the condition of independence.

How do I find the standard deviation of a linear combination of random variables?

To find the standard deviation of a linear combination of independent random variables
- take the positive square root of the variance
If $W = a X \pm b Y$ then $σ_{W} = \sqrt{a^{2} {σ_{X}}^{2} + b^{2} {σ_{Y}}^{2}}$

Examiner Tips and Tricks

Remember that $σ_{W} \neq |a| σ_{X} + |b| σ_{Y}$ .

Formulas for the standard deviation do not look nice because the square root function is not a linear operator, $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$ .

If you are asked to find the standard deviation of a linear combination, always find the variance first.

Is the distribution of adding two observations of X the same as the distribution of 2X?

If you add two independent observations of the random variable $X$ then this distribution is denoted $X_{1} + X_{2}$
- where $X_{1}$ is the random variable for the first observation
- and $X_{2}$ is the random variable for the second observation
- both $X_{1}$ and $X_{2}$ have the same distribution as $X$
If you double one observation of the random variable $X$ then this distribution is denoted $2 X$
The distribution of $X_{1} + X_{2}$ is not the same as the distribution of $2 X$
- The means of both are equal
  - $μ_{X_{1} + X_{2}} = μ_{2 X} = 2 μ_{X}$
- However, the variances are not equal
  - $σ_{X_{1} + X_{2}}^{2} = σ_{X_{1}}^{2} + σ_{X_{2}}^{2} = 2 σ_{X}^{2}$
  - $σ_{2 X}^{2} = 2^{2} σ_{X}^{2} = 4 σ_{X}^{2}$

Worked Example

An exam contains two sections, Section I and Section II, both of which are out of 100 points.

$X$ is the random variable for the number of points awarded to a randomly selected student in Section I and $Y$ is the random variable for the number of points awarded to that student in Section II.

The number of points awarded in each section can be assumed to be independent.

The mean and standard deviation of the two variables are given in the table.

Variable	Mean	Standard deviation
Number of points in Section I ( $X$ )	62.5	10.7
Number of points in Section II ( $Y$ )	74.1	8.4

The overall score is calculated by adding 40% of the number of points in Section I and 60% of the number of points in Section II.

Let the score of a randomly selected student be represented by the random variable $S$ . The score is calculated using the formula $S = 0.4 X + 0.6 Y$ .

(a) Calculate the expected value of $S$ .

Answer:

This is a linear combination of random variables

If $W = a X + b Y$ then $μ_{W} = a μ_{X} + b μ_{Y}$

Use the formula and the means in the table to calculate the mean of the scores

$\begin{array}{rcl} μ_{S} & = & 0.4 μ_{X} + 0.6 μ_{Y} \\ = & 0.4 (62.5) + 0.6 (74.1) \\ = & 69.46 \end{array}$

The expected value of $S$ is 69.46 points

(b) Calculate the standard deviation of $S$ .

Answer:

First, find the variance of $S$

If $W = a X + b Y$ then ${σ_{W}}^{2} = a^{2} {σ_{X}}^{2} + b^{2} {σ_{Y}}^{2}$

Use the formula and standard deviations in the table to calculate the variance of the scores

State the condition of independence

$X_{1}$ and $X_{2}$ are independent, therefore

$\begin{array}{rcl} σ_{S}^{2} & = & 0 . 4^{2} σ_{X}^{2} + 0 . 6^{2} σ_{Y}^{2} \\ = & 0 . 4^{2} \cdot 10 . 7^{2} + 0 . 6^{2} \cdot 8 . 4^{2} \\ = & 43.72 \end{array}$

Take the positive square root to find the standard deviation

$\begin{array}{rcl} σ_{S} & = & \sqrt{43.72} \\ = & 6.612 . . . \end{array}$

The standard deviation of $S$ is 6.61

Last updated: 13 August 2024

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Linear Combinations of Random Variables (College Board AP® Statistics)

Study Guide

Linear combinations of random variables

What is a linear combination of random variables?

How do I find the mean of a linear combination of random variables?

How do I find the variance of a linear combination of random variables?

Examiner Tips and Tricks

How do I find the standard deviation of a linear combination of random variables?

Examiner Tips and Tricks

Is the distribution of adding two observations of X the same as the distribution of 2X?

Worked Example

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Author: Dan Finlay

Author: Lucy Kirkham