## 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.

### Areas of Tails of Distributions

#### Key Takeaways

• The problem of finding the number $\mathrm{z}^{*}$ so that the probability $P\left(Z < \mathrm{z}^{*}\right)$ is a specified value $c$ is solved by looking for the number $c$ in the interior of Figure 12.2 "Cumulative Normal Probability" and reading $\mathrm{z}^{*}$ from the margins.
• The problem of finding the number $\mathrm{z}^{*}$ so that the probability $P\left(Z > \mathrm{z}^{*}\right)$ is a specified value $c$ is solved by looking for the complementary probability $1-c$ in the interior of Figure 12.2 "Cumulative Normal Probability" and reading $z^*$ from the margins.
• For a normal random variable $X$ with mean $\mu$ and standard deviation $\sigma$, the problem of finding the number $\mathrm{x}^{*}$ so that $P\left(X < \mathrm{x}^{*}\right)$ is a specified value $c$ (or so that $P\left(X > \mathrm{x}^{*}\right)$ is a specified value $c$) is solved in two steps: (1) solve the corresponding problem for $Z$ with the same value of $c$, thereby obtaining the $z$-score, $\mathrm{z}^{*}$, of $\mathrm{x}^{*}$; (2) find $\mathrm{x}^{*}$ using $\mathrm{x}^{*}=\mu+\mathrm{z}^{*} \cdot \sigma$.
• The value of $Z$ that cuts off a right tail of area $c$ in the standard normal distribution is denoted $z_{c}$.