A random variable attaches a number to each outcome. Its distribution lists every value with its probability.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The mean
The mean, or expected value, multiplies each value by its probability and adds. It is a long-run average.
Variance
Variance averages the squared deviations, weighted by probability. The standard deviation is its square root.
Linear transformations
Adding a constant shifts the mean and leaves the standard deviation alone. Multiplying scales both.
Combining variables
Means always add. Variances add only when the variables are independent, and they add for a difference too.
Standard deviations never add
Add the variances, then take the root. Adding standard deviations directly is the classic error here.
A random variable assigns numbers to outcomes
It turns a random process into a number, so that means and standard deviations become available. Discrete variables take separate values; continuous ones take any value in a range.
Expected value is a long-run average
The mean of a random variable is the average outcome over very many repetitions. It need not be an attainable value — an expected family size of 2.3 children is a perfectly sensible statistic.
Combining random variables
Means always add. Variances add only when the variables are independent, and they add for a difference as well as a sum — you never subtract variances. That is one of the most missed facts in the course.
Linear transformations
Adding a constant shifts the mean and leaves the standard deviation alone. Multiplying scales both. Knowing which measures shift and which scale explains why standardising works the way it does.
Step 2: Try It Yourself
Tap and try it out.
Total 4
Total 5
Total 6
Total 7
Total 7 has the most. It has 3 more than Total 4.
Step 3: Watch an Example
One step at a time.
Watch Marcus Combine Two Variables
X has mean 10 and standard deviation 3. Y has mean 4 and standard deviation 4, and the two are independent.
- Step 1
The mean of X + Y is 10 + 4 = 14, since means always add.
Step 4: Your Turn
Practice makes it stick.
The Sum
Problem 1 of 2
X has mean 10 and Y has mean 4. What is the mean of X + Y?
The Spread
Problem 2 of 2
Independent, with standard deviations 3 and 4. What is the standard deviation of the sum?
Mean and Spread
1 of 8
Values 0, 1, 2 with probabilities 0.25, 0.5, 0.25. Mean?
2 of 8
Standard deviations 6 and 8, independent. Standard deviation of the sum?
3 of 8
X has mean 20. What is the mean of X + 5?
4 of 8
X has standard deviation 4. What is the standard deviation of X + 5?
5 of 8
X has standard deviation 4. What is the standard deviation of 3X?
6 of 8
Independent with variances 9 and 16. Variance of the difference?
7 of 8
Sort each quantity by whether adding a constant changes it.
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- Empty
- Empty
8 of 8
Values 2 and 6 with probabilities 0.5 each. Mean?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Independent, standard deviations 5 and 12. Standard deviation of the sum?
Question 2 of 2
What happens to variances when independent variables are subtracted?
What You Learned
- The mean of a random variable is its expected value.
- Adding a constant shifts the mean and leaves the spread alone.
- Means always add; variances add when independent, and never standard deviations.