Random Variables

Mean and standard deviation of a distribution.

12 problems · about 22 minutes · VAR-5.A, VAR-5.D

What this lesson teaches

The student computes the mean and standard deviation of a discrete random variable and applies transformation rules.

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.

Straightforward practice. Get the method working first.

5 problems

Independent, standard deviations 5 and 12. Standard deviation of the sum?
Answer 13
Why 13.
What happens to variances when independent variables are subtracted?
Answer They add, exactly as for a sum.
Why Variances add for a difference too.
Values 0, 1, 2 with probabilities 0.25, 0.5, 0.25. Mean?
Answer 1
Why 0 + 0.5 + 0.5.
Standard deviations 6 and 8, independent. Standard deviation of the sum?
Answer 10
Why √(36 + 64).
X has mean 20. What is the mean of X + 5?
Answer 25
Why Adding shifts the mean.

The same ideas with more to keep track of.

3 problems

X has standard deviation 4. What is the standard deviation of X + 5?
Answer 4
Why Shifting does not change spread.
X has standard deviation 4. What is the standard deviation of 3X?
Answer 12
Why Multiplying scales it.
Independent with variances 9 and 16. Variance of the difference?
Answer 25
Why Variances add for a difference too.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each quantity by whether adding a constant changes it.
Answer Changes: The mean, The median · Unchanged: The standard deviation, The variance
Why Shifting moves the centre without changing the spread.
Values 2 and 6 with probabilities 0.5 each. Mean?
Answer 4
Why 1 + 3.
The Sum: X has mean 10 and Y has mean 4. What is the mean of X + Y?
Answer 14
Why 14.
The Spread: Independent, with standard deviations 3 and 4. What is the standard deviation of the sum?
Answer 5
Why 5.