Lecture6 - Probability Generating Function Lecture 6...

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1 Lecture 6 Probability Generating Function Definition: The probability generating function of a PMF; is defined as : Theorem: Let P(X=k) denote the PMF of a RV X with probability generating function (PGF) differentiable at Z=0. We then have k=0,… () ( ) x X P kP k == 0 { } Xk XX b Gz E z Pk z = x Gz 0 1 | ! x k X z k dG z kd z = ± Note: If we differentiate and let z=1,we recover moments of PMF. ± Or 1 11 |( ) | ( ) ( ) x xx x k zz dG z pkk z k pk E dz = ∑∑ 2 1 2 | ()( 1 ) | x x k p kkk z dz =− 2 (( 1 ) ) ( ) x x EE E ( 1 ) ' x x EG = ' ' 2 ( 1 ) ( 1 ) ( ( 1 ) ) ' x x Var G G G =+− Large Deviations and tail probabilities ± Computing probabilities for any distribution is generally difficult. ± Bounds on probabilities are in many practical cases sufficient. e.g. etc. Why are they useful? 9 Exact computation is avoided 9 Easier to evaluate . . . PX a β ≥<
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2 Markov Inequality ± Assume X is an RV in ,we can get a good estimate of a Prob. of X in some interval. ± Lemma : For a nonnegative RV X, with finite mean E(x) then we have [0, ) 0 a ∀> (x) () x E Pa a ≥≤ ± Proof: ± Or ( ) x X a f x d x ≥= 00 x a XX X a E xf x dx xf x dx xf x dx ∞∞ ==+ ∫∫ ( ) x aa xf xdx a f xdx aP a ≥≥ = x x E a Probability Bounds ± Note: The first Moment as expected provides an estimate of an order of probability. ± Chebyshev Inequality ± Variance is a measure of concentration of an RV near its mean . ± This implies that we can use such a quantity to estimate the likelihood of being in a neighborhood η (,) εη ε −+ ± Theorem: For any ± Since : ± Or 0 ε > (| | ) ( ) ( ) ηε P Xf x d x f x d x −∞ + −≥ = + || X X fx d x −≥ = 22 2 ( ) ( ) ση η X x d x x d x −∞ − ≥ =− ( | | ) ε εη ε X X d x PX 2 2 (| | ) σ
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Lecture6 - Probability Generating Function Lecture 6...

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