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Chapter6_ContinuousProbabilityDistributions

Chapter6_ContinuousProbabilityDistributions - Chapter 6...

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Chapter 6 Continuous Probability Distributions
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Continuous Probability Distribution Probability Distribution for r.v. X is…? Continuous random variable?
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Continuous Probability Distribution (continued) Requirement with probability dist? If X is continuous, then…
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Continuous Probability Distribution (continued) Consider the interval (a, b) where: - ≤ a ≤ b ≤ If a = b, then? The interval is a single point and P(single point)=0. We want prob. that X (a, b) P(a < X < b)… which is the area under the curve over the interval a to b.
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Normal Distribution aka Gaussian distribution Most useful and most commonly used Plays an important role in statistics b/c: Numerous phenomena seem to follow the normal dist. or can be approx. by it. Can be used to approx. discrete prob. dist. Plays very important role in classical statistical inference
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Normal Distribution (continued) Properties Normal curve is bell-shaped Mean Median Mode Perfectly symmetric about the mean (central value)
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Normal Distribution (continued) Characterized by: μ , the population mean σ , the population standard deviation Normal probability density function: x all for , e 2 1 ) x ( f 2 x 2 1 - - = σ μ π σ
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Normal Distribution (continued)
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Normal Distribution (continued) Function allows us to determine the probability that x falls in a given interval through numerical integration x all for , dx e 2 1 dx ) x ( f b a x 2 1 b a 2 - - = σ μ π σ
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Normal Distribution (continued)
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Normal Distribution (continued) Integration method difficult & time-consuming Pre-generated table would be appropriate Unfortunately, there are an infinite number of normal tables since there are an infinite number of combinations for μ and σ . Fortunately, ALL normal probability distributions can be represented in ONE table. Each value of X can be identified by the number of standard deviations it is away from the mean
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Normal Distribution & Empirical Rule
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Standard Normal Distribution Standard normal distribution (Z) has: mean ( μ ) equal to 0 standard deviation ( σ ) equal to 1 We write: Z ~ N(0, 1) which is read as “Z is normally distributed with μ = 0 and σ = 1.
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Standard Normal Distribution (continued) All normal probability distributions – any X~N( μ , σ ) – can be converted to a standard normal distribution Z~N(0,1) by standardization . If X~N( μ , σ ), X can be standardized to a z-score where Z~N(0,1) by σ μ - = X z
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Standard Normal Distribution (continued)
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Standard Normal Curve Gives cumulative probabilities P( Z < z) To use the table: Read the value of z down the leftmost column and across the top row Find the corresponding value of the probability in the body of the table
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Standard Normal Table Examples P(0 ≤ Z ≤ 1) P(Z ≤ 2.37) P(Z > -0.45) P(-1 ≤ Z ≤ 2.15) P(-1.6 ≤ Z ≤ 1.6) P(2 ≤ Z ≤ 3.5)
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Standard Normal Table Examples
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