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Unformatted text preview: Section 8.5 Normal Random Variables The Normal Distribution is sometimes referred to as the Gaussian Distribution after Carl Friedrich Gauss. The two names are interchangeable. We will try to only use “Normal” Distribution. Stirling’s Formula: .5 ! 2 m m m m e π + − ≈ when m is large. This formula provides the motivation for the Normal Distribution from the Binomial Distribution when n is large, and x is approximately equal to np (this is the first step, we also need to check when x is not = to np). PDF of the Normal Distribution 2 2 ( ) /2 1 ( ) 2 x X f x e µ σ πσ − − = x −∞ ≤ ≤ ∞ and σ > and µ is any real number µ is the mean of the Normal Distribution σ is the standard deviation of the Normal Distribution Surprisingly, we only need these 2 parameters and we know everything about the distribution. How do mean and standard deviation affect the shape of the graph? We are going to examine what happens when we change σ and hold µ constant and vice versa. σ is the standard deviation. What standard deviation measures is the spread from the mean. The following graph was created using 3 different Normal Distributions. They all have mean = 10. However, their standard deviations are 2, 4, and 9 respectively. The tallest one represents when σ is equal to 2. The shortest/widest represents when σ is equal to 9. The one in the middle is when σ is 4. Conclusion: Holding the mean constant, if we increase σ , we get a shorter/wider distribution. Holding the mean constant, if we decrease σ , we get a taller/narrower distribution (we usually say it is more peaked or more concentrated). What happens if we hold σ constant and change µ ? The following graph was created with 3 different Normal Distributions all of which having σ = 5. However, their means are 10, 20, and 30 respectively....
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 Spring '08
 MARTIN
 Normal Distribution, Standard Deviation, Probability theory, CDF

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