↑ Return to Random Variables

Normal

\[\]

\defl{Normal Distribution has the following properties:}

  • Symmetrical and a bell shaped appearance.
  • The population mean and median are equal.
  • An infinite range, $-\infty < x < \infty$
  • The approximate probability for certain ranges of $X$-values:
    1. $P(\mu - 1\sigma < X < \mu + 1\sigma) \approx 68\%$
    2. $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 95\%$
    3. $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 99.7\%$
\defs{Normal Distribution:}

\[ f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-\left(x-\mu\right)^2}{2\sigma^2}} \]

  • $-\infty < x < \infty$
  • $N(\mu,\sigma^2)$ is used to denote the distribution
  • $E(X) = \mu $
  • $V(X) = \sigma^2$
  • If $\mu=0$ and $\sigma^2=1$, this is called a Standard Normal and denoted $Z$
    • All Normally distributed random variables can be converted into a Standard Normal
  • To determine probabilities related to the Normal distribution the Standard Normal distribution is used
0.0.1. Standard Normal Distribution:The following formula is used to transform a Normally distributed random variable, $X$, into a Standard Normally distributed random variable,

\[ Z=\frac{X-\mu}{\sigma} \]

Note: if $z$ is known we can solve for $x$:

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

\[ f(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \]

  • $-\infty < z < \infty$
  • Has the same properties as the Normal distribution and
    • $N(0,1)$ is used to denote the distribution
    • $E(Z) = \mu=0 $
    • $V(Z) = \sigma^2=1$
  • The approximate probability for certain ranges of $Z$-values:
    1. $P(-1 < Z < 1) \approx 68\%$
    2. $P(-2 < Z < 2) \approx 95\%$
    3. $P(-3 < Z < 3) \approx 99.7\%$
  • To find probabilities of any $z$-value refer to the Table~sntable or computer software such as Microsoft Excel may be used
    1. $P(Z = c)=0$, where $c$ is a constant.
    2. $P(Z < -c)=P(Z > c)$
    3. $P(Z > c)=1-P(Z < c)$
    4. $P(Z < -c)=1-P(Z < c)$
0.0.2. The Distribution of the Average of I.I.D. Normally Distributed Random VariablesLet $X_1, X_2, \ldots, X_n$ be $n$ i.i.d. $N(\mu,\sigma^{2})$ random variables. The expectation of the sample mean, $\bar{X}$, of $n$ i.i.d. Normally distributed random variables is $\mu$ and the variance is $\frac{\sigma^{2}}{n}$. Recall equations eofxbar and vofxbar. The situation where the random variables are Normally distributed is a special case in that $\bar{X}$ is also Normally distributed and

\[\bar{X} \sim N(\mu,\frac{\sigma^{2}}{n}). \]

In addition,

\[ Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}, \]

where $Z\sim N(0,1)$. \defs{Examples}
  • The height of a randomly selected person is often assumed to be Normally distributed
  • The weight of a randomly selected person is often assumed to be Normally distributed
  • The pulse rate of a randomly selected person is often assumed to be Normally distributed