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Chen iecuhk df x dx engg2430c lecture 8 22 26 cdf

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Unformatted text preview: x |y ) = fY |X (y |x ) fY (y ) pX |Y (x |y ) = pY |X (y |x ) Total probability law pX (x ) = ∑ pX ,Y (x , y ) = ∑ pY |X (y |x )pX (x ) y ˆ fX (x ) = y fY |X (y |x )fX (x ) dy fX ,Y (x , y ) dy = y M. Chen ([email protected]) ˆ y ENGG2430C lecture 8 21 / 26 CDF – Cumulative Distribution Function CDF of a continuous r.v. X , denoted by FX (x ), is defined as ˆx FX (x ) = P (X ≤ x ) = fX (x ) dt −∞ (From the textbook) Relationship between CDF and PDF fX ( x ) = M. Chen ([email protected]) dF (x ) dx ENGG2430C lecture 8 22 / 26 CDF – Exponential R.V. Let X be an exponential r.v. with parameter λ . Its PDF is fX (x ) = Its CDF is FX ( x ) λ e −λ x , 0, ˆ = P (X ≤ x ) = x if x ≥ 0; otherwise. λ e −λ z dz = 1 − e −λ x ∀x ≥ 0 0 It is straightforward to verify that f X (x ) = M. Chen ([email protected]) dF (x ) = dx λ e −λ x , 0, ENGG2430C lecture 8 if x ≥ 0; otherwise. 23 / 26 CDF – Cumulative Distribution Function CDF of a discrete r.v. X , denoted by FX (x ), is defined as F X (x ) = P (X ≤ x ) = ∑ pX ( k ) k ≤x (From the textbook) FX (x ) is an increasing function. That is, if x ≤ y , then FX (x ) ≤ FX (y ). F (−∞) =?, F (∞) =? M. Chen ([email protected]) ENGG2430C lecture 8 24 / 26 CDF Application: Exp. and Geo. RVs Geometric r.v. approaches Exponential r.v. if we fix λ , let δ goes to 0, but fix p = 1 − e −λ δ ≈ λ δ . Consider time is chopped into slots with length δ . In each slot, we flip a coin and get a head with probability p and a tail with probability 1 − p .We have t Fgeo (t ) = 1 − P X > = 1 − (1 − p ) δ ˆt Fexp (t ) = λ e −λ τ dt = 1 − e −λ t t δ 0 Let δ goes to 0 and observe p ≈ λ δ , we have Fgeo (t ) ≈ 1 − (1 − λ δ ) M. Chen ([email protected]) t δ ≈ 1 − e −λ δ ENGG2430C lecture 8 t δ ≈ 1 − e −λ t = Fexp (t ). 25 / 26 Thank You Reading: Ch. 3 of the textbook. Next lecture: Derived Distributions, Generating Random Variables and Inequalities. M. Chen ([email protected]) ENGG2430C lecture 8 26 / 26...
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