Lecture4 - Properties of a PDF Lemma: 1. f X ( x) is a...

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1 ECE 514 Lecture 4 Properties of a PDF ± Lemma: 1. is a nonnegative function , i.e. , 2. integrates to 1. Note : () X f x () 0 X fx x R ∀∈ X f x 1 X f xdx = lim ( ) ( ) 1 y XX yy F yf x d x −∞ →∞ = = ± Question: How do we use PDF’s to compute Probabilities? Lemma : Recalling that We have the following: Question: Does satisfy Axiom 3 of Probability? x F xf y d y = ( ) b X a Pa X b f xdx + + <≤= ( ) b X a + ≤≤= ( ) b X a ≤<= ( ) b X a + <<= ( ) X a Pa x + <= X f Conditional PDF ± Consider a probability space and a random variable defined on . Definition: The conditional probability density function of an RV , given some event A such that is defined as: {, ,} P Ω F Ω X PA (| ) X X dF x A fxA dx =
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2 Empirical Distribution Ex: Suppose you are measuring a voltage V 1 which is supposed to be constant. ± As a sensed voltage, it feeds a regulator(filter) whose output needs to maintain a constant temperature to within a ( for example) Problem: To design a regulator we will need the statistics (variations of ) i.e. which we need to establish experimentally. 1 Va = 1 10 D 1 V 1 ii n =+ 1,. .. iT = (Continued) Question: How would we go about that? Start with Definition : Or: () ( ) X P Xxd xP Xx fx dx +− ( ) X f xdx Px X x dx ≤≤+ ± i.e. Find the mass in that fractional segment.
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Lecture4 - Properties of a PDF Lemma: 1. f X ( x) is a...

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