lect13 - CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM...

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CONTINUOUS RANDOM VARIABLES Outline CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM VARIABLES EXPECTATION THE NORMAL DISTRIBUTION 1 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal
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CONTINUOUS RANDOM VARIABLES CONTINUOUS RANDOM VARIABLES A random variable X is said to have a continuous distribution if there exists a nonnegative function f such that P ( X B ) = R B f ( x ) dx for any set B of real numbers. In particular, R -∞ f ( x ) dx = . Interpretation of f : if f is continuous at a point x , then lim Δ 0 P ( X [ x , x + Δ]) Δ = lim Δ 0 R x x f ( y ) dy Δ = . Shorthand for this: P ( x X x + dx ) = f ( x ) dx . f is called the probability density function (for short, pdf or just density) of X . The function F : ( -∞ , ) [ 0 , 1 ] defined by F ( x ) = P ( X x ) is called the cumulative distribution function (cdf for short) of X . 2 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal
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CONTINUOUS RANDOM VARIABLES RELATIONSHIP BETWEEN DENSITY AND CDF If X has density f , then you get its cdf by integration : F ( x ) = P ( X ( -∞ , x ]) = Z x -∞ f ( y ) dy . Conversely, if the cdf F of X is differentiable, or if F is everywhere continuous and differentiable except at finitely many points, then by the fundamental theorem of calculus , P ( X ( x , z ]) = F ( z ) - F ( x ) = Z x z F 0 ( y ) dy for all -∞ < x < z < , so X has density f = . 3 / 16 Xinghua Zheng Lect 13: Continuous random variables; uniform and normal
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CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM VARIABLES A random variable X is said to be uniformly distributed over the interval [ a , b ] , written X Uniform ( a , b ) , if it has a density f X which is constant on that interval, and equal to 0 outside that interval: c = What is the graph of the cdf F X of X ? For a α < β b , P ( α < X β ) = . Put U = ( X - a ) / ( b - a ) . For 0 u 1, the cdf of U is F U ( u ) = P ± X - a b - a u ² = P ( a X a + u ( b - a )) = , so U has density f U ( u ) = F 0 U ( u ) = ( , if 0 u 1 ; , otherwise .
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lect13 - CONTINUOUS RANDOM VARIABLES UNIFORM RANDOM...

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