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A brief introduction to the calculus of continuous random variables.

A random variable is a quantity whose value depends on the outcome of some random event or process:
• When rolling a six-sided die, there is a random variable $X$ associated to this action which represents the side of the die that comes up on top. In this case, possible values of $X$ are $1,2,3,4,5,$ or $6$, and (if it’s a fair die) each outcome is equally likely. This is an example of a discrete random variable, which means that the possible values are finite or otherwise clearly separated from one another.
• For any given brand of lightbulb, there is an associated random variable $T$ which represents the lifetime of the bulb (i.e., $T$ equals the total time that a particular bulb lasts before it burns out). We idealize this random variable to be a continuous random variable, because the lifetime could be any positive amount of time in principle (i.e., we might have one bulb lasting $1000.0423$ hours and another lasting only $10^{-9}$ seconds longer than that).

We will focus primarily on continuous random variables. The information we will typically need to know in order to work with continuous random variables are:

• The interval $I$ of possible values that the random variable may take. For the lightbulb burn-out example, this would be the interval $[0,\infty )$.
• A probability density function to compute the probability of various events.
As the name suggests, a PDF $f(x)$ represents a density of probability; so if $f(x)$ is relatively large at a particular value of $x$, this indicates that values near that $x$ are relatively more likely to occur than values near where $f(x)$ is smaller.