Lets say that expected value is defined as follows: Let X be a random variable, ie. X can take a value from a set of real numbers, U = {R1, R2, R3, ... ,Rn} in a random manner.
Example: let X be the number when a 6-sided die is rolled. Each time the die is rolled, X takes one value from the set U = {1, 2, 3, 4, 5, 6}.
Further suppose that X takes value xi with probability pi. To be valid probabilities, each pi is a real number such that pi is between 0 and 1 inclusive, and they must sum to unity:(sum of all probability is 1)
We say that P(X = xi) = pi, that is, the probability that X takes the value xi is pi. Example: for a fair 6-sided die, P(X = xi) = 1/6, for xi is a member of the set U.
suppose that I have a fair 6-sided die, the expected value E[X] would be E[X] = (1/6)(1 + 2 + 3 + 4 + 5 + 6) = 3.5
what if i have a biased (unfair) 6-sided die, and let X be the number when it is rolled once. The probabilities are given by: P(X = xi) = ki, for i = 1,2.... 6, where k is an unknown constant.
K would refer to the outcome for each possible value of the unfair die right? How would I find K so that i can find the expected value of this unfair die?
Guide: