STAXXXX_Notes_ExamThree

STAXXXX_Notes_ExamThree - Created by Dane McGuckian...

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Created by Dane McGuckian Continuous Probability Distributions In this section we work with random variables that are continuous. A continuous random variable can assume any numerical value within some interval or intervals. Let X be just such a continuous random variable, then: The probability function ( ) f x for X, a continuous random variable, is called a probability density function (pdf). s A probability density function must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x -axis.) The area beneath the curve ( ) f x between two points (a) and (b) is equal to the probability that the random variable x takes on value between (a) and (b). That is: ( ) P a x b < < = A = the area between points (a) and (b) under the curve ( ) f x . We use calculus normally to find these areas; however, in this course we will use tables or software to find the areas/probabilities. There is an important property regarding areas of this type that we should cover. Since a continuous random variable can take on an infinite, uncountable number of values there is zero probability that x will take on any specific value. In other words, P(x = a) = 0 for any a in the function’s domain. This means that unlike discrete distributions, P( a < x < b) = ( ) ( ) ( ) P a x b P a x b P a x b ≤ ≤ = < ≤ = ≤ < Example 74 Male weights are normally distributed with a mean of 172 lbs and a standard deviation of 29 lbs. What is the probability that a randomly selected man from the population will weigh exactly 182 lbs?
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Created by Dane McGuckian Solution: P( x = 182) = 0, since weights are normally distributed, we will have zero probability of selecting a male who weighs exactly 182 lbs. The Uniform Distribution s X can take on any value between c and d with equal probability = 1/ ( d - c ) s For two values a and b ( ) b a P a x b d c c a b d - < < = - ≤ < ≤ The Normal Distribution One of the most common continuous distributions is the Normal distribution . It is bell-shaped, so we call it the bell curve. It is perfectly symmetrical around its mean. Many natural phenomena can be modeled by the Normal distribution. The Normal Probability Distribution ( ) ( ) ( ) 2 2 / 2 1 2 x f x e μ σ π - - = where x -∞ < < ∞ Note: 2.7183 e and 3.1416 Since many different random variables follow a Normal or approximately Normal distribution (each combination of μ and σ produces a unique normal curve). We could create a table to give us areas from under the Normal curve, but we would need a different table for each random variable because their scales would all be different (different means and standard deviations). To fix this problem we can standardize our random variables: The standardized normal random variable x z - =
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Created by Dane McGuckian The mean of z is 0, and the standard deviation is 1. The standard normal distribution has three important properties:
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This note was uploaded on 12/11/2011 for the course STA 2122 taught by Professor Staff during the Fall '08 term at FIU.

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STAXXXX_Notes_ExamThree - Created by Dane McGuckian...

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