Mean for a DPD:

\[\mu=\sum_{i=1}^nx_i\cdot P(x_i)\]

Variance for a DPD:

\[\Big(\sum_{i=1}^nx_i^2\cdot P(x_i)\Big)-\mu^2\]

Standard Deviation for a DPD:

\[\sigma=\sqrt{\sigma^2}=\sqrt{\Big(\sum_{i=1}^n x_i^2\cdot P(x_i)\Big)-\mu^2}\]

Finding \(z\)-score:

\[z=\frac{x-\mu}{\sigma}\]

Finding normal value from \(z\)-score:

\[x=z\cdot\sigma+\mu\]

Mean of the sample means:

\[\mu_{\overline{x}}=\mu\]

Standard Error of the mean:

\[\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}\]

Converting \(\overline{x}\) to \(z\)-scores:

\[z=\frac{\overline{x}-\mu}{\Big(\frac{\sigma}{\sqrt{n}}\Big)}=\frac{\sqrt{n}\big(\overline{x}-\mu\big)}{\sigma}\]