Expected Value & Variance Rules Cheat Sheet

A printable reference covering expected value, variance, linear transformations, sums, independence, binomial distributions, and normal distributions for grades 10-12.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Expected value and variance are two of the most important tools for describing random variables. Expected value gives the long-run average outcome, while variance measures how spread out the outcomes are. This cheat sheet helps students quickly choose the correct rule for single random variables, transformed variables, sums, and common distributions. It is especially useful for probability, statistics, and data analysis problems in grades 10-12. The core ideas are that expected value behaves like an average and follows linear rules, while variance follows different rules because it measures squared distance from the mean. For a random variable $X$ , the mean is $E(X)$ and the variance is $Var(X)=E\left[(X-\mu)^2\right]$ . Adding constants changes expected value but not variance, while multiplying by a constant multiplies variance by the square of that constant. For independent random variables, expected values add and variances add.

Key Facts

For a discrete random variable, the expected value is $E(X)=\sum xP(X=x)$ .
Variance can be calculated with $Var(X)=E\left[(X-\mu)^2\right]$ , where $\mu=E(X)$ .
The shortcut variance formula is $Var(X)=E(X^2)-\left[E(X)\right]^2$ .
For a linear transformation, $E(aX+b)=aE(X)+b$ .
For a linear transformation, $Var(aX+b)=a^2Var(X)$ , so adding $b$ does not change the variance.
For any two random variables, $E(X+Y)=E(X)+E(Y)$ .
If $X$ and $Y$ are independent, then $Var(X+Y)=Var(X)+Var(Y)$ and $Var(X-Y)=Var(X)+Var(Y)$ .
For a binomial random variable $X\sim Bin(n,p)$ , $E(X)=np$ and $Var(X)=np(1-p)$ .

Vocabulary

Random variable: A variable whose value is determined by the outcome of a random process.
Expected value: The long-run average value of a random variable over many repetitions, written as $E(X)$ .
Variance: A measure of spread equal to the average squared distance from the mean, written as $Var(X)$ .
Standard deviation: The square root of variance, written as $\sigma=\sqrt{Var(X)}$ , and measured in the original units.
Independent random variables: Random variables where knowing the value of one does not change the probability distribution of the other.
Linear transformation: A change to a random variable in the form $aX+b$ , where $a$ stretches or reflects values and $b$ shifts values.

Common Mistakes to Avoid

Adding a constant to variance, such as writing $Var(X+5)=Var(X)+5$ , is wrong because shifting all values does not change their spread.
Forgetting to square the multiplier in variance, such as writing $Var(3X)=3Var(X)$ , is wrong because variance uses squared distances, so $Var(3X)=9Var(X)$ .
Subtracting variances for a difference, such as writing $Var(X-Y)=Var(X)-Var(Y)$ , is wrong for independent variables because spreads add, so $Var(X-Y)=Var(X)+Var(Y)$ .
Using $E(X^2)=\left[E(X)\right]^2$ is wrong because the expected square is usually not the square of the expected value.
Adding variances without checking independence is risky because $Var(X+Y)=Var(X)+Var(Y)$ only works when $X$ and $Y$ are independent.

Practice Questions

1 A game pays $0$ dollars with probability $0.5$ , $4$ dollars with probability $0.3$ , and $10$ dollars with probability $0.2$ . Find $E(X)$ .
2 If $E(X)=12$ and $Var(X)=9$ , find $E(2X-5)$ and $Var(2X-5)$ .
3 Let $X\sim Bin(20,0.3)$ . Find $E(X)$ and $Var(X)$ .
4 A teacher says that subtracting two independent random variables should subtract their variances. Explain why this is incorrect using the meaning of variance.

Sign in to save