Study Notes on Normal Distribution and Standard Normal Distribution

Normal Distribution

Introduction to Normal Distribution

• The Normal Distribution is defined by the probability density function (PDF):
  $f(x) = \frac{1}{\sigma imes ext{sqrt{2 ext{π}}}} e^{-\frac{(x - ext{µ})^2}{2 ext{σ}^2}}$
• The parameters:
  • µ (mu): The mean of the distribution
  • σ (sigma): The standard deviation of the distribution
• By varying parameters µ and σ, one achieves different normal distributions, referred to as a family of Normal curves.

Historical Context

• The Normal Distribution formulation was first associated with Thomas Simpson, in connection with measurement errors in astronomical observations.
• The concept was later expanded by Carl Friedrich Gauss, a German mathematician.
• In some countries, including Germany, the Normal Distribution is known as the Gaussian Distribution.

Characteristics of Normal Distribution

• Shape: Bell-shaped curve that is symmetric about its mean.
• Equal Measures: The Mean, Median, and Mode of a normal distribution are all equal.
• Determinants:
  • Location is determined by the mean, µ.
  • Spread is determined by the standard deviation, σ.
• The random variable's theoretical range is infinite, ranging from $- ext{∞}$ to $+ ext{∞}$.

Properties of Normal Distribution

1. Symmetrical about the mean
2. Fully defined by mean (µ) and standard deviation (σ)
3. Total area under the curve equals 1 (100%)
4. The x-axis is a horizontal asymptote for both tails of the distribution.

Shape and Shifting of Distribution

• Change in μ shifts the curve left or right.
• Change in σ affects the spread of the distribution.
• Notation for Normal Distribution:
  • $X ilde N( ext{µ}, ext{σ})$

Area Under the Curve

• The total area under the curve is equal to 1.
• Probabilities can be interpreted as areas under the curve, where:
  • $P(- ext{∞} < X < + ext{∞}) = 1$
  • $P( ext{µ} < X < + ext{∞}) = 0.5$
  • $P(- ext{∞} < X < ext{µ}) = 0.5$

Standard Normal Distribution

• A standard normal distribution is achieved through the process of standardizing any normal variable X:
  • Standardized score/z-score defined as:
    $Z = \frac{X - ext{µ}}{ ext{σ}}$
• The standard normal distribution is expressed as:
  • $Z ilde N(0,1)$

Example of Standardization

• If $X$ has a distribution of $ext{µ} = 10$ and $ext{σ} = 5$, then for an x-value of 20:
  $Z = \frac{20 - 10}{5} = 2$
• This indicates that x = 20 is two standard deviations above the mean.

Comparing Units of X and Z

• Original units (X) and standardized units (Z) can be expressed/compared using:
  • Original: $ext{µ} = 100, ext{σ} = 50$
  • Standardized: $ext{µ} = 0, ext{σ} = 1$

Finding Probabilities in Normal Distribution

For an Interval [a, b]

• The probability for the interval $P(a < X < b)$ can be identified as:
  $P(a < X < b) = P(Z < b) - P(Z < a)$

Example Calculations

1. Example with probability:

   • Given $P(Z > 2.00) = 0.0228$
   • Calculation of upper tail probabilities:
     • $P(Z > 0.59) = 1 - P(Z ext{≤} 0.59) = 1 - 0.7224 = 0.2776$

2. For the interval $P(-0.76 < Z < 1.12)$, use standard values in Z-table:

   • $P(Z < -0.76) = 0.2236$
   • $P(Z < 1.12) = 0.8686$
   • Hence:
   • $P(-0.76 < Z < 1.12) = 0.8686 - 0.2236 = 0.6450$

3. Z-values for different probabilities:

   • For a known probability of $P(Z > c) = 0.1093$, determine:
   • Z ext{ that cuts off 10.93% on the right side: Z = 1.23}

4. For the lower 20%:

   • $Z = -0.84$
   • Resulting value:
     $X = ext{µ} + ext{σ} imes Z = 8 + (5 imes -0.84) = 3.80$

Finding X Values from Known Probabilities

1. To find $X$ for given probabilities:

   • Step 1: Find the corresponding $Z$ value for the known probability.
   • Step 2: Convert back to X using the formula:
   • $X = ext{µ} + ext{σ} imes Z$

2. Example with mean 73 and standard deviation 9.2:

   • To find the X value corresponding to the 90th percentile (90% below):
     • Find $Z$ value of 1.28.
     • Then:
     • $X = 73 + (9.2 imes 1.28)$
     • Resulting in $X = 84.78$

Conclusion

• The Normal Distribution is essential in statistics and is widely applied in real-world scenarios involving data approximations. Understanding probabilities and how to compute them using Z-scores facilitates easier analysis of continuous random variables.