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 , the mean is and the variance is . 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 .
- Variance can be calculated with , where .
- The shortcut variance formula is .
- For a linear transformation, .
- For a linear transformation, , so adding does not change the variance.
- For any two random variables, .
- If and are independent, then and .
- For a binomial random variable , and .
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 .
- Variance
- A measure of spread equal to the average squared distance from the mean, written as .
- Standard deviation
- The square root of variance, written as , 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 , where stretches or reflects values and shifts values.
Common Mistakes to Avoid
- Adding a constant to variance, such as writing , is wrong because shifting all values does not change their spread.
- Forgetting to square the multiplier in variance, such as writing , is wrong because variance uses squared distances, so .
- Subtracting variances for a difference, such as writing , is wrong for independent variables because spreads add, so .
- Using is wrong because the expected square is usually not the square of the expected value.
- Adding variances without checking independence is risky because only works when and are independent.
Practice Questions
- 1 A game pays dollars with probability , dollars with probability , and dollars with probability . Find .
- 2 If and , find and .
- 3 Let . Find and .
- 4 A teacher says that subtracting two independent random variables should subtract their variances. Explain why this is incorrect using the meaning of variance.
Understanding Expected Value & Variance Rules
Variance is built from deviations, which are the distances between outcomes and the mean. Those distances are squared before they are averaged. Squaring does two important jobs.
It prevents negative and positive deviations from cancelling out. It gives unusually distant outcomes much more influence. This is why a variable with a few extreme values can have a large variance even when most outcomes sit near the center.
The unit of variance is squared too. If a measurement is in centimetres, its variance is in square centimetres.
Standard deviation is often easier to interpret because it returns to the original unit. It is the square root of variance.
A useful way to understand transformations is to think about changing a scoring scale. Adding ten points to every test score moves the whole distribution upward by ten points. Each student remains the same distance from every other student, so the spread does not change.
Multiplying every score by two doubles every distance from the mean. Since variance uses squared distances, the variance becomes four times as large.
This square effect is a common source of mistakes. A negative multiplier reverses the order of values, but it still has the same effect on spread as the matching positive multiplier.
Expected values can be combined even when two quantities are related. This makes them reliable for planning totals. For example, a shop can add the expected sales from two products to estimate expected total sales.
Variance needs more care. Independence means that knowing one result gives no information about the other. When quantities are independent, their random ups and downs do not tend to move together, so the spread rules for sums and differences work cleanly.
When they are not independent, covariance matters. Positively related quantities make a total more variable.
Negatively related quantities can make a total less variable. Students should never assume independence simply because two variables have different names.
Binomial models appear in repeated yes or no settings, such as counting made free throws, defective parts, or people who answer correctly. The model needs a fixed number of trials, only two outcome categories, the same success probability on each trial, and independence between trials. Real situations may only approximately meet these conditions.
Drawing items without replacement can change the probability from draw to draw. For large numbers of trials, a binomial distribution often has a shape close to a normal distribution when successes and failures are both sufficiently common.
That approximation helps with probability estimates, but it is less dependable near zero or near the total number of trials. Always identify the random variable in words first, then check its conditions before choosing a rule.