A random variable is a variable that whose value can change under different outcomes. For example, the result of flipping a standard 6-sided die is a random variable that takes each of the values from 1 to 6 with probability
A random variable contains a lot of information. In Expected Value, we learned that the weighted average of all possible outcomes gives one way of understanding the random variable. Recall that .
The variance of a random variable measures how far the random variable deviates from its mean, by calculating the expected value of the square deviation from the mean. In mathematical symbols, we have
It can also be shown, by the linearity of expectation, that
1. There are bags, containing balls numbered through . From each bag, ball is removed. What is the variance of the total of the two balls?
Let be the random variable denoting the sum of these values. Then, the probability distribution of is given by
We have previously calculated that As such, we can see that
2. six-sided dice are rolled. What is the variance for the number of times a is rolled? What is the variance for the total of the dice?
Let be the random variable representing the number of times a is rolled. The below table lists the probabilities of rolling different numbers of s.
The expected number of times a is rolled is , so . We now wish to calculate .