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  • Special Distributions

Standard Normal Distribution

STA256

• This is a special case of the Normal Distribution.
• $\mu =0$ and ${\sigma }^2=1$
• Often associated with the letter $\mathcal{Z}$
• Cumulative Probabilities under the Standard Normal are denoted with $\mathrm{\Phi }(\mathcal{Z})$
• $\mathrm{\Phi }(\mathcal{Z})=P(\mathcal{Z}\le z)=F(z)$
• $\mathrm{\Phi }(\mathcal{Z})$ is the area to the left of $z$
• $\mathrm{\Phi }(\mathcal{Z})={\int }_{-\mathrm{\infty }}^z\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}\text{differential}x$

STA258

• Ex:
  • $\mathrm{\Phi }(0)=\frac{1}{2}$
  • $\mathrm{\Phi }(-Z)=1-\mathrm{\Phi }(Z)$

• (2. Descriptive Statistics, p.64)

  • $P(0<Z<1.82)$
  • $\mathrm{\Phi }(0)-\mathrm{\Phi }(1.82)$
  • $0.5-0.344=0.156=15.6\mathrm{\%}$

• • $P(-1.82<Z<0)$
  • By symmetry, same as above.

• • $P(Z\le 0.73)=1-P(Z>0.73)$
  • $=1-0.2327=0.7673$

• • $P(-2\le Z\le 2)=P(Z\ge -2)-P(Z.=2)=1-P(Z\ge 2)-P(Z\ge 2)=1-2P(Z\ge 2)$
  • $=1-(2)(0.0228)=0.9544$

• • $P(Z\ge z_0)=0.0158$
  • $z_0=2.15$