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**Normal Probability Distribution**

**Definition : Probability Density Function**

A probability density function is an equation used to compute probabilities of continuous random variables that must satisfy the following two properties.

- Graph of the equation must be greater than or equal to zero for all possible values of the random variable.
- Area under the curve equals 1.

**EXAMPLE : Illustrating the Uniform Distribution **

Imagine that a friend of yours is always late. Let the random variable X represent the time from when you are supposed to meet your friend until he shows up. Further suppose that your friend could be on time (x = 0) or up to 30 minutes late (x = 30) with all 1 – minute intervals of times between x = 0 and x = 30 equally likely. That is to say, your friend is just as likely to be from 3 to 4 minutes late as he is to be 25 to 26 minutes late. The random variable X can be any value in the interval from 0 to 30, that is, 0 ≤ X ≤ 30. Because any two intervals of equal length between 0 and 30, inclusive, are equally likely, the random variable X is said to follow a **uniform probability distribution.**

**Properties of the Normal Probability Curve : **

- The highest point occurs at x = µ.
- It is symmetric about the mean, µ. One half of the curve is a mirror image of the other half, i.e., the area under the curve to the right of µ is equal to the area under the curve to the left of µ equals ½.
- It has inflection points at µ – σ and µ + σ.
- The curve is asymptotic to the horizontal axis at the extremes.
- The total area under the curve equals one.

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**Properties of the Normal Probability Curve (continued) : **

- Empirical Rule :

- Approximately 68 % of the area under the curve is between µ – σ and µ + σ.
- Approximately 95 % of the area under the curve is between µ – 2σ and µ + 2σ.
- Approximately 99.7 % of the area under the curve is between µ – 3σ and µ + 3σ.

A normal curve has two characteristics : mean (µ) and standard deviation (σ).

**Example 1 –** normal curves for two populations with different means :

**Summary :** The two curves are exactly the same, except one curve is to the right of the other curve.

**Example 2 –** normal curves for two populations with different standard deviations.

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**Summary :** Increasing the standard deviation causes the curve for Population #2 to become flatter and more spread out. Comparing the two normal curves :

- For Population #1, there is more area under the curve within a given distance of the mean;
- For Population #2, there is more area under the curve away from the mean.

**Standardized Variable –** A variable is said to be standardized if it has been adjusted (or transformed) such that its mean equals 0 and its standard deviation equals 1.

The z – score represents the number of standard deviations that a data value is away from the mean.

**Normal Probability Distributions (or Curves).**

- A normal curve is characterized by its mean, µ, and standard deviation, σ.
- Since there are an infinite number of combinations of µ’s and σ’s, there are likewise an infinite number of normal curves.
- One particular type of normal curve is the standard normal curve…a normal curve with µ = 0 and σ = 1.

**The Standard Normal Distribution**

Standardizing a Normal Random Variable

Suppose the random variable X is normally distributed with mean µ and standard deviation σ. Then the random variable

is normally distributed with mean µ = 0 and standard deviation σ = 1. The random variable Z is said to have the standard normal distribution.

**Standard Normal Distribution (Z)**

**Properties of the Standard Normal Curve (Z) : **

- The highest point occurs at µ = 0.
- It is a bell – shaped curve that is symmetric about the mean, µ = 0. One half of the curve is a mirror image of the other half, i.e., the area under the curve to the right of µ = 0 is equal to the area under the curve to the left of µ = 0 equals ½.
- It has inflection points at µ – σ = 0 – 1 = – 1 and µ + σ = 0 + 1 = + 1.
- The curve is asymptotic to the horizontal axis at the extremes.
- The total area under the curve equals one.
- Empirical Rule :

- Approximately 68 % of the area under the curve is between – 1 and + 1.
- Approximately 95 % of the area under the curve is between – 2 and + 2.
- Approximately 99.7 % of the area under the curve is between – 3 and + 3.

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