Question

Can you provide examples of when there are more than 2 potential outcomes for a random variable and it is mutually exclusive? What is the relationship between a random variable being mutually exclusive and the sum of all the random variables being 1 in a probability distribution?

Answer 1

Mutually exclusive concept can be explained based on events not on random variable. A die throwing experiment is a classic example of more than two potential outcomes where the events defined under the random variable X assuming 6 values are mutually exclusive. ie happening of one on the die will not include happening of 2 or 3 or 4or 5or 6. In the case of a discrete random variable

Each value of a random variable defines an event and a random variable will cover all the events of an experiment eg: suppose if we define another random variable Y =0 as the event of having even numbers on the die, then Y=1 will represent the compliment of the event or it may even represent a single value eg: the appearance of 3; In that case The event {1,5} will be represented by another value say 2 for Y.The union of all the events defined under Y are exhaustive that is the union has covered all the possible outcomes of the die throwing experiment. So in that sense we say that any random varible is mutually exclusive or the events defined under the values assumed by the random variables are mutually exclusive. There fore the probability distribution of a random variable will have individual probabilities for the happening of the events defined under the value of random variable.The sum of such probabilities of values of a particular random variable is always 1. If the events are mutually exclusive and exhastive then the total probability of occurance of all the mutually exclusive events is 1