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 )
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
FX (x ) = P (X ≤ x ) =
fX (x ) dt
−∞ (From the textbook) Relationship between CDF and PDF
fX ( x ) =
M. Chen ([email protected]) dF
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 ,
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;
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
Fgeo (t ) = 1 − P X >
= 1 − (1 − p )
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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- Spring '14
- Probability theory, M. Chen, Minghua Chen, engg2430c lecture