The Normal Distribution and Other Continuous Distributions

The Normal Distribution and Other Continuous Distributions

Continuous Probability Distributions

Continuous probability distributions describe the behavior of continuous random variables, which can take on any value within a given range. These distributions are fundamental in business statistics for modeling real-world phenomena.

• Continuous variable: A variable that can assume any value on a continuum (an uncountable number of values).

• Examples:

  • Thickness of an item

  • Time required to complete a task

  • Temperature of a solution

  • Height, in inches

• These variables can potentially take on any value, limited only by the precision and accuracy of measurement.

The Normal Distribution

The normal distribution is one of the most important continuous probability distributions in statistics, especially in business applications.

• Shape: Bell-shaped and symmetrical.

• Central Tendency: The mean, median, and mode are all equal.

• Parameters:

  • Mean (μ): Determines the location of the center of the distribution.

  • Standard deviation (σ): Determines the spread (width) of the distribution.

• Range: The random variable has an infinite theoretical range: from $-\infty$ to $+\infty$.

The Standardized Normal Distribution (Z Distribution)

Any normal distribution can be transformed into the standardized normal distribution, also known as the Z distribution. This transformation allows for easier calculation of probabilities and comparison between different normal distributions.

• Standardized normal distribution (Z): A normal distribution with a mean of 0 and a standard deviation of 1.

• To compute normal probabilities, transform X units into Z units.

Translation to the Standardized Normal Distribution

To convert a value from a normal distribution (X) to the standardized normal distribution (Z), use the following formula:

• Formula:

$Z = \frac{X - \mu}{\sigma}$

• The Z distribution always has mean = 0 and standard deviation = 1.

Properties of the Standardized Normal Distribution

• Also known as the "Z" distribution.

• Mean is 0.

• Standard deviation is 1.

• Values above the mean have positive Z-values.

• Values below the mean have negative Z-values.

Example: Calculating a Z-Score

If X is distributed normally with a mean of 100 and a standard deviation of 50, the Z value for X = 200 is:

$Z = \frac{X - \mu}{\sigma} = \frac{200 - 100}{50} = 2.0$

• This means that X = 200 is two standard deviations above the mean of 100.

Comparing X and Z Units

• The shape of the distribution remains the same after standardization; only the scale changes.

• Problems can be expressed in the original units (X) or in standardized units (Z).

Finding Normal Probabilities

Probabilities in a normal distribution are measured by the area under the curve.

• Total area under the curve: 1.0 (or 100%)

• The curve is symmetric, so half the area is above the mean and half is below.

• Probability of any individual value is zero; probabilities are for intervals.

The Cumulative Standardized Normal Table

The cumulative standardized normal table (often found in textbook appendices) provides the probability that a standard normal variable Z is less than a given value.

• Rows: Show the value of Z to the first decimal point.

• Columns: Show the value of Z to the second decimal point.

• Table value: Gives the probability from Z = $-\infty$ up to the desired Z.

General Procedure for Finding Normal Probabilities

1. Draw the normal curve for the problem in terms of X.

2. Translate X-values to Z-values using the standardization formula.

3. Use the cumulative standardized normal table to find the required probability.

Example: Finding Normal Probabilities

Suppose X represents the time (in seconds) to download an image file from the internet. X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds. Find $P(X < 18.6)$.

• Standardize: $Z = \frac{18.6 - 18.0}{5.0} = 0.12$

• Look up $P(Z < 0.12)$ in the table: $P(Z < 0.12) = 0.5478$

Finding Upper Tail Probabilities

To find $P(X > 18.6)$:

• $P(X > 18.6) = P(Z > 0.12) = 1.0 - P(Z < 0.12) = 1.0 - 0.5478 = 0.4522$

Finding a Normal Probability Between Two Values

Suppose X is normal with mean 18.0 and standard deviation 5.0. Find $P(18 < X < 18.6)$.

• Standardize: $Z_1 = \frac{18 - 18.0}{5.0} = 0$, $Z_2 = \frac{18.6 - 18.0}{5.0} = 0.12$

• $P(0 < Z < 0.12) = P(Z < 0.12) - P(Z < 0) = 0.5478 - 0.5000 = 0.0478$

Probabilities in the Lower Tail

Suppose X is normal with mean 18.0 and standard deviation 5.0. Find $P(17.4 < X < 18)$.

• Standardize: $Z_1 = \frac{17.4 - 18.0}{5.0} = -0.12$, $Z_2 = 0$

• $P(-0.12 < Z < 0) = P(Z < 0) - P(Z < -0.12) = 0.5000 - 0.4522 = 0.0478$

• Because the normal distribution is symmetric, this probability is the same as $P(0 < Z < 0.12)$.

Given a Normal Probability, Find the X Value

To find the X value for a known probability:

1. Find the Z value corresponding to the known probability (using the cumulative standardized normal table).

2. Convert to X units using the formula: $X = \mu + Z\sigma$

Example: Finding the X Value for a Known Probability

Suppose X is normal with mean 18.0 and standard deviation 5.0. Find X such that 20% of download times are less than X.

• Find the Z value for 20% in the lower tail: $Z = -0.84$ (from the table, $P(Z < -0.84) = 0.2000$)

• Convert to X: $X = 18.0 + (-0.84)(5.0) = 18.0 - 4.2 = 13.8$

Step	Description	Formula
Standardize X	Convert X to Z	$Z = \frac{X - \mu}{\sigma}$
Find Probability	Use Z to find area under curve	Table lookup or $P(Z < z)$
Find X for Probability	Given probability, find Z, then X	$X = \mu + Z\sigma$