# Continuous Random Variables

## 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 $z_{c}$.