Chapter6_ContinuousProbabilityDistributions

Chapter6_ContinuousProbabilityDistributions - Chapter 6...

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Unformatted text preview: Chapter 6 Continuous Probability Distributions Continuous Probability Distribution Probability Distribution for r.v. X is? Continuous random variable? Continuous Probability Distribution (continued) Requirement with probability dist? If X is continuous, then 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. 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 Normal Distribution (continued) Properties Normal curve is bell-shaped Mean Median Mode Perfectly symmetric about the mean (central value) Normal Distribution (continued) Characterized by: , the population mean , the population standard deviation Normal probability density function: x all for , e 2 1 ) x ( f 2 x 2 1 -- = Normal Distribution (continued) 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 -- = Normal Distribution (continued) 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 and . 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 Normal Distribution & Empirical Rule Standard Normal Distribution Standard normal distribution (Z) has: mean ( ) equal to 0 standard deviation ( ) equal to 1 We write: Z ~ N(0, 1) which is read as Z is normally distributed with = 0 and = 1. Standard Normal Distribution (continued) All normal probability distributions any X~N( , ) can be converted to a standard normal distribution Z~N(0,1) by standardization . If X~N( , ), X can be standardized to a z-score where Z~N(0,1) by - = X z Standard Normal Distribution (continued) 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 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) Standard Normal Table Examples Standard Normal Table Examples Standard Normal Table Examples Normal Distribution...
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Chapter6_ContinuousProbabilityDistributions - Chapter 6...

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