Unformatted text preview: P (a ≤ X ≤ b ) = fX (x ) dx
ˆa a P (a ≤ X ≤ a ) = fX (x ) dx = 0
a ´∞ −∞ fX (x ) dx =1 (The δ trick) If δ is very small, then
P (X ∈ [x , x + δ ]) ≈ fX (x )δ
M. Chen ([email protected]) ENGG2430C lecture 8 12 / 26 PDF Can Take Arbitrary Value True or false? Let X be a continuous r.v. and fX (x ) is its PDF, then
0 ≤ fX (x ) ≤ 1.
False. For example, consider a uniform r.v. taking values in a small
range.
PDF is density of probability mass, and it can take arbitrary values. A
´
valid PDF only requires P (X ∈ B ) = B fX (x ) dx ≤ 1, ∀B ⊆ R .
Example: Consider a random variable X with PDF
fX (x ) = 1
√,
2x if 0 < x ≤ 1; 0, otherwise. This PDF can become inﬁnity as x goes to 0. But it is still a valid
PDF, because fX (x ) ≥ 0 and
ˆ1
√
fX (x )dx = x 1 = 1
0
0 M. Chen ([email protected]) ENGG2430C lecture 8 13 / 26 PDF of Discrete Random Variable Let X be a Bernoulli r.v. that takes values 1 or 0, and
pX (1) = p
pX (0) = 1 − p
The plot of its PDF is as follows M. Chen ([email protected]) ENGG2430C lecture 8 14 / 26 Exponential R. V. Let X be an exponential r.v. with PDF fX (x ), then
fX ( x ) = λ e −λ x ,
0, if x ≥ 0;
otherwise. (ﬁgure from wiki)
The time interval between two packet arrivals at a router
The life time of a bulb
M. Chen ([email protected]) ENGG2430C lecture 8 15 / 26 Exponential r. v. – Memoryless Property * Let X be an exponential r.v.
ˆ ∞ P (X ≥ a) =
ˆ λ e −λ x dx = e −λ a ∀a ≥ 0 a
∞ x λ e −λ x dx
0
ˆ
−λ x ∞
(integration by parts) = (−xe
)0+
E [X ] = ∞ e −λ x dx = 1/λ 0 Var(X ) = E [X 2 ] − E 2 [X ] = 1/λ 2
Given X ≥ a, ﬁnd fX {X ≥a} (x ).
fX {X ≥a} (x ) =
M. Chen ([email protected]) λ e −λ (x ...
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This document was uploaded on 03/31/2014.
 Spring '14

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