## Continuous Random Variables

First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using $z$-score transformations. Finally, it defines tail probabilities and illustrates how to find them.

### Continuous Random Variables

#### Key Takeaways

• For a continuous random variable $X$ the only probabilities that are computed are those of $X$ taking a value in a specified interval.
• The probability that $X$ take a value in a particular interval is the same whether or not the endpoints of the interval are included.
• The probability $P(a < X < b)$, that $X$ take a value in the interval from $a$ to $b$, is the area of the region between the vertical lines through $a$ and $b$, above the $x$-axis, and below the graph of a function $f(x)$ called the density function.
• A normally distributed random variable is one whose density function is a bell curve.
• Every bell curve is symmetric about its mean and lies everywhere above the $x$-axis, which it approaches asymptotically (arbitrarily closely without touching).