Normal Distributions

Normal Distributions - 4.3 NORMAL PROBABILITY DISTRIBUTIONS...

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4.3 NORMAL PROBABILITY DISTRIBUTIONS The Most Important Probability Distribution in Statistics

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A random variable X with mean μ and standard deviation σ is normally distributed if its probability density function is given by ... 71828 . 2 e and ... 14159 . 3 where x e 2 1 ) x ( f 2 x ) 2 / 1 ( = = π - π σ = σ μ - - Normal Distribution
The Shape of the Normal Distribution The normal distribution is bell shaped, and symmetrical around μ. μ y symmetrical? Let μ = 100. Suppose x = 110. 2 2 10 ) 2 / 1 ( 100 110 ) 2 / 1 ( e 2 1 e 2 1 ) 110 ( f σ - σ - - π σ = π σ = Now suppose x = 90 2 2 10 ) 2 / 1 ( 100 90 ) 2 / 1 ( e 2 1 e 2 1 ) 90 ( f σ - - σ - - π σ = π σ = 110 90

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Normal Probability Distributions The expected value (also called the mean ) μ can be any number The standard deviation σ can be any nonnegative number There are infinitely many normal distributions
The effects of The effects of μ μ and and σ σ How does the standard deviation affect the shape of f(x)? σ = 2 σ =3 σ =4 μ = 10 μ = 11 μ = 12 How does the expected value affect the location of f(x)?

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X 8 3 6 9 12 0 μ A family of bell-shaped curves that differ only in their means and standard deviations. µ = the mean of the distribution σ = the standard deviation µ = 3 and σ = 1
X 3 6 9 12 0 X 3 6 9 12 0 μ µ = 3 and σ = 1 μ µ = 6 and σ = 1

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X 8 3 6 9 12 0 μ X 8 3 6 9 12 0 μ µ = 6 and σ = 2 µ = 6 and σ = 1
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Normal Distributions - 4.3 NORMAL PROBABILITY DISTRIBUTIONS...

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