2 3 as as is a nondecreasing function of 5 6

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Unformatted text preview: er x as follows 1. 2. 3. as as is a nondecreasing function of 5 6 Properties of the pdf Probability Density Function (pdf) • Continuous random variable • Nonnegative function f defined on the real line. • For every real interval 1. 2. fX(x) 3. • First two properties follow from the axioms of probability. • Integrate: x x x + dx P x X x dx f X ( x)dx 7 8 Example: Spin the Pointer Uniform Distribution 1 , a xb f X ( x ) b a 0, elsewhere 1 1/(b a) a x1 x2 b 2 2 1 x • Probability of value in any subinterval of is proportional to its length. • Area of rectangle must be unity. 1 2 2 1 MATLAB >> rand(m, n) % a=0, b=1, m by n 9 10 Expectation of a Random Variable Density & Distribution • Expected value or mean of . • Justified by relative frequency • Discrete . • Continuous 11 12 Properties of Expectation Function of a Random Variable • Expected value of the function • Discrete: • Continuous: 13 14 Moments Variance moment: expectation of power • First moment is the mean • Second moment is the mean square • Discrete • Second moment about the mean • Continuous • Continuous Standard Deviation: square root of variance • Discrete 15 16 Variance Property: Proof Properties of the Variance = − = + = + = • Uncorrelated: − − − = ( ), − 17 Example: Uniform Distribution 18 Characteristic Function Find the mean and variance = →...
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