# 5: Continuous Random Variables and The Normal Distribution

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Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.

• 5.1: Continuous Probability Functions
The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a<x<b)P(a<x<b) . The cumulative distribution function (cdf) gives the probability as an area.
• 5.2: Prelude to The Normal Distribution
The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them. The normal distribution has two parameters (two numerical descriptive measures), the mean (μ) and the standard deviation (σ).
• 5.3: The Standard Normal Distribution
A z-score is a standardized value. Its distribution is the standard normal, Z∼N(0,1). The mean of the z-scores is zero and the standard deviation is one. If y is the z-score for a value x from the normal distribution N(μ,σ) then z tells you how many standard deviations x is above (greater than) or below (less than) μ.
• 5.4: Using the Normal Distribution
The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean μ and the standard deviation σ. A special normal distribution, called the standard normal distribution is the distribution of z-scores. Its mean is zero, and its standard deviation is one.
• 5.5: The Uniform Distribution
The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.
• 5.6: The Exponential Distribution
The exponential distribution is often concerned with the amount of time until some specific event occurs. Values for an exponential random variable occur in the following way. There are fewer large values and more small values. The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.
• 5.7: Continuous Distribution (Worksheet)
A statistics Worksheet: The student will compare and contrast empirical data from a random number generator with the uniform distribution.
• 5.8: Normal Distribution - Lap Times (Worksheet)
A statistics Worksheet: The student will compare and contrast empirical data and a theoretical distribution to determine if Terry Vogel's lap times fit a continuous distribution.
• 5.9: Normal Distribution - Pinkie Length (Worksheet)
A statistics Worksheet: The student will compare empirical data and a theoretical distribution to determine if data from the experiment follow a continuous distribution.
• 5.E: The Normal Distribution (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

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