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An introduction to continuous probability distributions, specifically the uniform and normal distributions. It covers the shape, probability density functions, mean, standard deviation, and calculating probabilities using standard normal distributions. Examples and tables are included to illustrate the concepts.

Typology: Study notes

Pre 2010

1 / 4

Download Probability Distributions: Understanding Uniform and Normal Distributions and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! 9/24/2007 1 Chapter 5 Continuous Random Variables Continuous Probability Distributions Continuous Probability Distribution – areas under curve correspond to probabilities for x Area A corresponds to the probability that x lies between a and b 2 Do you see the similarity in shape between the continuous and discrete probability distributions? The Uniform Distribution Uniform Probability Distribution – distribution resulting when a continuous random variable is evenly di t ib t d ti l 3 s r u e over a par cu ar interval ( ) cd xf − = 1 Probability Distribution for a Uniform Random Variable x Probability density function: Mean: Standard Deviation: dxc ≤≤ 2 dc + =μ 12 cd − =σ ( ) ( ) ( ) dbaccdabbxaP ≤<≤−−=<< ,/ The Normal Distribution A normal random variable has a probability distribution called a normal distribution 4 The Normal Distribution Bell-shaped curve Symmetrical about its mean μ Spread determined by the value of it’s standard deviation σ The Normal Distribution The mean and standard deviation affect the flatness and center of the curve, but not the basic shape 5 The Normal Distribution The function that generates a normal curve is of the form where ( ) ( ) ( )[ ]221 2 1 σμ πσ −−= xexf 6 μ = Mean of the normal random variable x σ = Standard deviation π = 3.1416… e = 2.71828… P(x<a) is obtained from a table of normal probabilities 9/24/2007 2 The Normal Distribution Probabilities associated with values or ranges of a random variable correspond to areas under the normal curve Calculating probabilities can be simplified by working with a Standard Normal Distribution 7 A Standard Normal Distribution is a Normal distribution with μ =0 and σ =1 The standard normal random variable is denoted by the symbol z The Normal Distribution Table for Standard Normal Distribution contains probability for the area between 0 and z Partial table shows components of table 8 The Normal Distribution What is P(-1.33 < z < 1.33)? Table gives us area A1 Symmetry about the mean 9 tell us that A2 = A1 P(-1.33 < z < 1.33) = P(-1.33 < z < 0) +P(0 < z < 1.33)= A2 + A1 = .4082 + .4082 = .8164 The Normal Distribution What is P(z > 1.64)? Table gives us area A2 Symmetry about the mean 10 tell us that A2 + A1 = .5 P(z > 1.64) = A1 = .5 – A2=.5 - .4495 = .0505 The Normal Distribution What is P(z < .67)? Table gives us area A1 Symmetry about the mean 11 tell us that A2 = .5 P(z < .67) = A1 + A2 = .2486 + .5 = .7486 The Normal Distribution What is P(|z| > 1.96)? Table gives us area .5 - A2 =.4750, so A2 = .0250 12 Symmetry about the mean tell us that A2 = A1 P(|z| > 1.96) = A1 + A2 = .0250 + .0250 =.05