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Summary 5

# Summary 5 - Chapter 5 summary Continuous Random Variables...

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Chapter 5 summary Continuous Random Variables Definition of a continuous random variable: A random variable called continuous (it is also called absolutely continuous) if there exists a function f X with the property that for any two numbers a, b with a < b , the probability that X lies between a and b is P ( a < X < b ) = -∞ f X ( x ) dx. The function f X is called the probability density function of X , or just the density function of X . It follows from this that for any a , P ( X = a ) = 0, so the above probability is the same as P ( a X b ). General properties of a probability density function: f X ( x ) 0 for all x R , and -∞ f X ( x ) dx = 1 . Relation between the distribution and the density of a continuous random variable: The distribution is still defined to be F X ( x ) = P ( X X ). We have F X = f X , and F X ( x ) = x -∞ f x ( t ) dt. Expected value of a continuous random variable: E [ X ] = -∞ x f X ( x ) dx. Expected value of a function of a continuous random variable: E [ g ( X )] = -∞ g ( x ) f X ( x ) dx. (1) Variance and standard deviation of a continuous random variable: The variance of X is defined to be V ar ( X ) = E [( X - E ( X )) 2 ] = -∞ ( x - E ( X )) 2 f X ( x ) dx, but it is usually calculated using the simpler formula V ar ( X ) = E [ X

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