Continuous Random Variables

Areas of Tails of Distributions

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 \(z_{c}\).