# Random variable, an outcome, an event, mutually exclusive and exhaustive events

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Random variable, an outcome, an event, mutually exclusive and exhaustive events

A random variable is a variable whose outcome is uncertain, and you have no control over its outcome. Suppose you toss a fair coin. Two results are possible- head or tail. Either of those two outcomes can occur, and only one of those two outcomes will occur. We don't know the outcome and also don't have the control over it. An outcome is an observed value of a random variable.

An event is a specified set of outcomes. The event can have a single outcome or multiple outcomes. For example, when we throw a dice then an event stating that the number is 5 is an event with a single outcome. An event stating that the number is less than 5 has multiple outcomes (1,2,3, and 4).

The mutually exclusive events are events that cannot happen simultaneously. The events A = the portfolio earns a return below 8 percent and B = the portfolio earns a return above 8 percent are mutually exclusive events, and both events cannot occur together at the same time.

The exhaustive events cover the range of all possible outcomes of an event. The events A and B discussed above are not exhaustive events because they are missing an outcome of a portfolio return of 8 percent. If we add an event C = the portfolio earns a return of 8 percent, then events A, B, and C become exhaustive events as they cover the entire range of returns possible.

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