# LL1 - P(X = a = P(a X a = Part 1 Continuous Random...

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BUS1200-1 1 Part 1 Continuous Random Variables Definition Suppose X is a random variable. If there exists a non-negative function f( x ) such that P( a X b ) = b a x x d ) f( holds for all real numbers a and b with a b , then X is called a continuous random variable and f( x ) is called a probability density function of the random variable X . Therefore, if X is a continuous random variable with f( x ) being a probability density function, then Probability Density Function f( x ) a b 0 x BUS1200-1 2 P( X = a ) = P( a X a ) = a a x x d ) f( = 0 for any real number a ; that is, the probability of taking on any particular value is 0. Furthermore, we have P( a < X < b ) = P( a < X b ) = P( a X < b ) = P( a X b ) = the area under the graph of f( x ) from a to b , and + x x d ) f( = 1 where the left hand side is the area between the whole graph of f( x ) and the x axis. Definition The expectation, expected value or mean of a continuous random variable X is defined as + x x x d ) f( , (if it exists) and is denoted by E( X ) or μ , where f( x ) is a probability density function of X . If µ = E( X ) exists, the variance of X is defined as BUS1200-1 3 + x x x d ) f( ) ( 2 (if it exists) and is denoted by Var( X ) or σ 2 .

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LL1 - P(X = a = P(a X a = Part 1 Continuous Random...

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