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Continuous Probability Distributions Study Guide (page 2)

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Updated on Oct 5, 2011

Normal Distribution

Normal Probability Distributions

Normal probability distributions are continuous probability distributions that are bell shaped and symmetric. They are also known as Gaussian distributions or bell-shaped curves.

The normal distribution is perhaps the most widely used probability distribution, largely because it provides a reasonable approximation to the distribution of many random variables. It also plays a central role in many of the statistical methods that will be discussed in later lessons. Normal probability distributions are continuous probability distributions that are bell shaped and symmetric as displayed in Figure 11.5. The distribution is also called the Gaussian distribution or the bell-shaped curve.

Figure 11.5

The normal distribution has two parameters: the mean μ and the standard deviation σ. The notation X ~ N(μ ,σ) means that "X is normally distributed with a mean of μ and a standard deviation of σ". The distribution is symmetric about the mean. The mean, median, and mode are all equal. The mean is often referred to as the location parameter because it determines where the distribution is centered. The standard deviation determines the spread of the distribution. The effect of the mean and standard deviation on the normal distribution is displayed in Figure 11.6.

Figure 11.6

For any normal distribution, about 68% of the observations are within one standard deviation of the mean. About 95% and 99.7% of the observations are, respectively, within two and three standard deviations of the mean.

It is important to remember that, although the location and spread may change, the area under the curve and above the x-axis is always 1. Unfortunately, the probabilities associated with intervals cannot be computed easily as with the uniform distribution. To overcome this difficulty, we rely on a table of areas for a reference of normal distribution called the standard normal distribution. The standard normal distribution is the normal distribution with μ = 0 and σ = 1. It is customary to use the letter z to represent a standard normal random variable.

We will first learn to compute probabilities for a standard normal random variable and then learn how to find them for any random variable. We will also want to be able to determine extreme values of z, such as the value that only 5% of the population exceeds or the value that 1% of the population is less than. To find either probabilities or extreme values, we need a table of standard normal curve areas, or we need a calculator or computer that can be used to find these values. Here, we will restrict ourselves to the use of tables. The standard normal table used here in Table 11.1 tabulates the probability of observing a value less than or equal to z (see Figure 11.7).

Table 11.1 Standard normal probabilities, P(z < z*)

Figure 11.7

Graphs are extremely useful tools to help us understand what values we are searching for. We will do this for each problem we work.

 

Examples of Continuous Probability Distributions

Below are nine examples of continuous probability distributions problems and solutions.

Example 1

Find P(z < 1.32).

Solution 1

Using the standard normal table, we find the row with 1.3 in the z column and move along that row to the 0.02 column to find 0.9066. Thus, P(z < 1.32) = 0.9066. Figure 11.8 shows the graphic image of this.

Figure 11.8

 

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