Continuous Probability Distribution: Chapter 6

Continuous Probability Distribution: Chapter 6

• The lecture covers continuous probability distributions within a software application context.
• Rocelle Ann G. Terco is the facilitator for course AE 09 (Lec).

Learning Objectives

• Define and illustrate normal random variables, describing their characteristics.
• Grasp the properties of normally distributed random variable graphs.
• Define and illustrate standard normal distributions.
• Transform a normal distribution into a standard normal distribution using a z-score.
• Find areas under the standard normal curve by using a standard normal table.

Concept Map

• A random variable may be described by a probability distribution; Properties and types will be explored.
• Properties include expected value and variance while types may be discrete or continuous.
• Probability distribution falls into discrete and continuous distributions which is further classified as Probability Mass Function or Normal Distribution.

Normal Distribution Basics

• A normal or Gaussian distribution appears as a bell-shaped graph.
• It relies on two parameters: mean and standard deviation.
• The graph of a normal distribution is called the normal curve.

Examples of Normal Distribution

• Height, rolling a dice, tossing a coin, IQ and Shoe Size
• Birth weight.

Properties of a Normal Distribution

• The mean, median, and mode are equal in a normal distribution.
• The normal curve has a bell shape and is symmetric around the mean.
• The total area under the normal curve is equal to 1.
• The normal curve approaches, but never touches, the x-axis as it extends away from the mean.
• Inflection points are where the curve changes from curving upward to curving downward.
• Between 𝜇−𝜎 and 𝜇+𝜎 , the graph curves downward.
• The graph curves upward to the left of 𝜇−𝜎 and to the right of 𝜇+𝜎 .

Empirical Rule

• Approximately 68% of the area under the curve falls within one standard deviation from the mean.
• About 95% of the area is within two standard deviations from the mean.
• Approximately 99.7% of the area is within three standard deviations from the mean.

Graph Factors

• The graph of a normal distribution hinges on the mean (𝜇) and standard deviation (𝜎).
• The mean dictates the center's position.
• A change in the mean's value shifts the normal curve left or right.
• Standard deviation dictates the shape of the graph, specifically its height and width.
• A large standard deviation yields a short and wide curve.
• A small standard deviation yields a skinnier and taller graph.

Interpreting Graphs

• The scaled test scores for the New York State Grade 8 Mathematics Test are normally distributed.
• The mean test score is estimated to be approximately 675.
• The standard deviation is estimated to be approximately 35.

Introduction to Standard Normal Distribution

• Transforming each data value of a normally distributed random variable x into a z-score yields the standard normal distribution.
• Standardizing helps compare normal distributions with varying spread, position, or units.
• Normal distributions can have any mean and standard deviation with units denoted by x, the area can be found using the empirical rule.
• Standard normal distribution has a fixed mean of 0 and a standard deviation of 1 with units denoted by z, while the area can be found using the standard normal table.
• The normal distribution with a mean of 0 and a standard deviation of 1 is the standard normal distribution.

Z-Scores

• Z-scores on the horizontal scale indicate how many standard deviations a value is from the mean.
• The cumulative area is close to 0 for z-scores near z = -3.49.
• The cumulative area increases as z-scores increase.
• The cumulative area for z = 0 is 0.5000.
• The cumulative area is close to 1 for z-scores near z = 3.49.

Finding Area with Standard Normal Table

• Locate the first two digits (tenth value) of the z-score in the column.
• Find the last digit (hundredth value) along the uppermost row.
• The intersection of the row and column provides the area, only half of the normal curve.
• The reference point is the Mean, so the maximum area is 50% or 0.5.