Normal Distribution

• The most important continuous probability distribution in the entire field of statistics is the normal distribution.
• The normal curve describes approximately many phenomena that occur in nature, industry and research (human height, measurement errors, stock market!, etc.).
• In 1733, Abraham DeMoivre developed the mathematical equation of the normal curve.
• The normal distribution is often referred to as the Gaussian distribution, in honour of Karl Friedrich Gauss (1777-1855), who also derived its equation from a study of errors in repeated measurements of the same quantity.
• The term normal distribution is a historical accident because there is nothing particularly normal about the normal distribution and nor is there anything abnormal about other distribution.

• Normal Distribution:
  

  Figure 2: The normal curve.

  

  A continuous random variable $X$ having the bell-shaped distribution of Fig. 2 is called a normal random variable.

• Notes:
  • $(x-\mu)^2$ is squared distance from the mean
  • $e^{-\frac{1}{2\sigma^2}(x-\mu)^2}$ get smaller as $(x-\mu)^2$ gets larger
  • How fast it gets small depends on $\sigma$. Faster for small $\sigma$.
  • The term $\frac{1}{\sqrt{2\pi \sigma}}$ makes sure $\int_{-\infty}^\infty n(x;\mu,\sigma)dx=1$

Figure 3: Normal curves with $\mu _1 < \mu _2$ and $\sigma _1 = \sigma _2$.

Figure 4: Normal curves with $\mu _1 =\mu _2$ and $\sigma _1 < \sigma _2$.

Figure 5: Normal curves with $\mu _1 < \mu _2$ and $\sigma _1 < \sigma _2$.

The properties of the normal curve

1. The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at $x =\mu$.
2. The curve is symmetric about a vertical axis through the mean $\mu$.
3. The curve has its points of inflection at $x =\mu \pm \sigma$ , is concave downward if $\mu-\sigma<X<\mu+\sigma$, and is concave upward otherwise.
4. The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean.
5. The total area under the curve and above the horizontal axis is equal to 1.
6. Both tails become dramatically thin beyond $\pm 3\sigma$ from the mean $\mu$.

• A certain type of battery lasts on average3 years with a standard deviation of 0.5 years.
• Assuming battery lives are normally distributed,
• Find the probability that a given battery will last less than 2.3 years;
• Solution:
  $$P(X < 2.3)= \int_{-\infty}^{2.3}\frac{1}{\sqrt{2\pi 0.5}}e^{-\frac{1}{2*0.5^2}(x-3)^2}dx$$

• Difficult to solve!
• Then, tabulation of normal curve areas is necessary.