Discussion 7.pdf - Discussion 7 Continuous Probability Distributions De\u2026nition 1 Probability density function(pdf If X is a continuous random variable

Discussion 7.pdf - Discussion 7 Continuous Probability...

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Discussion 7 Continuous Probability Distributions De°nition 1 Probability density function (pdf)- If X is a continuous random variable, then a function f ( x ) such that for any two numbers a and b with a ° b; we get P ( a ° X ° b ) = R b a f ( x ) dx: This means that the probability that X takes on a value in the interval [ a; b ] is the area above the interval and under the density function. Criteria in order for f ( x ) to be a legitimate pdf: 1. f ( x ) > 0 for all x: 2. R 1 °1 f ( x ) dx = 1 ; the total are of the function must be 1 : De°nition 2 Cumulative distribution function (cdf)- If X is a continuous random variable, then F ( x ) is de°ned by F ( x ) = P ( X ° x ) = R x °1 f ( y ) dy: For each x; F ( x ) is the are under the density curve to the left of x: De°nition 3 Uniform distribution- A continuous random variable X has a uniform distribution on [ A; B ] if the pdf is given by f ( x ) = ° 1 B ° A ; A ° x ° B 0 ; otherwise 1
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Main Concepts: 1. P ( a ° X ° b ) = P ( a < X < b ) = P ( a < X ° b ) = P ( a ° X < b ) 2. P ( X > a ) = 1 ± F ( a ) 3. P ( a ° X ° b ) = F ( b ) ± F ( a ) 4. If X is a continuous random variable with pdf f ( x ) and cdf F ( x ) ; then F 0 ( x ) = f ( x ) for
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  • Fall '08
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  • Normal Distribution, Probability distribution, Probability theory, probability density function, CDF

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