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Lecture2014Ch7Fa07

# Lecture2014Ch7Fa07 - Chapter 7 The Normal Probability...

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Chapter 7 – The Normal Probability Distribution The normal distribution is a special type of bell-shaped curve. Defn : A random variable X is said to be normally distributed or to have a normal distribution if its distribution has the shape of a normal curve: ( 29 ( 29 2 2 2 2 1 σ μ σ π - - = x e x f , for - < x < , - < μ < , and σ > 0. Here μ and σ are the parameters of the distribution; μ = the mean of the random variable X (or of the probability distribution); and σ = the standard deviation of X. The two mathematical constants appearing in the formula, π 2245 3.141592654, and e 2245 2.718281828. The normal distribution is not just a single distribution, but rather a family of distributions; each member of the family is characterized by a particular pair of values of μ and σ . The mean, μ , tells us the location of the center of the bell-shaped curve. The standard deviation, σ , tells us how wide the bell-shaped curve is; smaller values of σ denote narrower, more sharply peaked bell-shaped curves; larger values of σ denote wider, less sharply peaked bell-shaped curves. In particular, the points given by μ - σ and μ + σ are the inflection points of the curve. Examples of Normal Distributions:

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Characteristics of normal distributions : 1) The bell is symmetric around the mean value μ . 2) The standard deviation, σ , describes the width of the bell. 3) The distribution is unimodal (one peak), and the mean, median and mode are all equal. 4) The curve is continuous; i.e., there are no gaps. 5) For any two numbers, a and b, the probability that the random variable X will be found to have a value between a and b is denoted by P(a X
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Lecture2014Ch7Fa07 - Chapter 7 The Normal Probability...

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