Chen iecuhk engg2430c lecture 8 12 26 pdf can take

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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 (IE@CUHK) 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 (IE@CUHK) 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 (IE@CUHK) 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 (IE@CUHK) 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 (IE@CUHK) λ e −λ (x...
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