Probability

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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.

A probability density function $f$ on an interval $[a,b]$ is a function which is

(a): Non-negative, i.e., $f(x) \geq 0$ for every $x$ in the interval $[a,b]$ ,
(b): Can be integrated on the interval $[a,b]$ ; this is possible if, for example, $f$ is continuous on $[a,b]$ ,
(c): Has total integral $1$ , i.e., $\int _a^b f(x) dx = 1.$ (Note: we can allow $a = -\infty$ and/or $b = \infty$ if these make sense for a particular random variable. In this case, we interpret the integral as an improper integral.)

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.

Suppose $X$ is a random variable taking values between $a$ and $b$ which has a probability density function $f(x)$ on the interval $[a,b]$ . Then for any two numbers $c \leq d$ inside the interval $[a,b]$ , the probability that $X$ takes a value between $c$ and $d$ is given by integration: $P( c \leq X \leq d) = \int _c^d f(x) dx.$

When plotting a PDF, the horizontal axis ranges over possible values of the random variable $X$ and the vertical axis measures relatively likelihood. The area under the graph equals the probability that $X$ takes values between $c$ and $d$ .