ReviewPart2

# ReviewPart2 - 1 Probability Density Functions(PDF For a...

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Unformatted text preview: 1 Probability Density Functions (PDF) For a continuous RV X with PDF f X ( x ) ( ≥ 0), P ( a ≤ X ≤ b ) = integraldisplay b a f X ( x ) dx P ( x ≤ X ≤ x + δ ) ≈ f X ( x ) .δ P ( X ∈ A ) = integraldisplay A f X ( x ) dx Remarks:- if X is continuous, P ( X = x ) = 0 ∀ x !!- f X ( x ) may take values larger than 1. Normalization property: integraldisplay ∞-∞ f X ( x ) dx = 1 2 2 Mean and variance of a continuous RV E [ X ] = integraldisplay ∞-∞ xf X ( x ) dx E [ g ( X )] = integraldisplay ∞-∞ g ( x ) f X ( x ) dx Var( X ) = integraldisplay ∞-∞ ( x- E [ X ]) 2 f X ( x ) dx = E [ X 2 ]- ( E [ X ]) 2 ( ≥ 0) E [ aX + b ] = a E [ X ] + b Var( aX + b ) = a 2 Var( X ) 3 3 Cumulative Distribution Functions Definition: F X ( x ) = P ( X ≤ x ) monotonically increasing from 0 (at-∞ ) to 1 (at + ∞ ). • Continuous RV: F X ( x ) = P ( X ≤ x ) = integraldisplay x-∞ f X ( t ) dt (continuous) f X ( x ) = dF X dx ( x ) • Discrete RV: F X ( x ) = P ( X ≤ x ) = summationdisplay k ≤ x p X ( k ) (piecewise constant) p X ( k ) = F X ( k )- F X ( k- 1) 4 4 Normal/Gaussian Random Variables Standard Normal RV: N (0 , 1): f X ( x ) = 1 √ 2 π e- x 2 / 2 E [ X ] = 0 , Var( X ) = 1 General normal RV: N ( μ,σ 2 ): f X ( x ) = 1 σ √ 2 π e- ( x- μ ) 2 / 2 σ 2 E [ X ] = μ, Var( X ) = σ 2 5 • if Y = aX + b , then Y ∼ N ( aμ + b, a 2 σ 2 ). • CDF for standard normal φ ( . ) can be read in a table. • To evaluate CDF of a general standard normal, express it as a function of a standard normal: X ∼ N ( μ,σ 2 ) ⇔ X- μ σ ∼ N (0 , 1) P ( X ≤ x ) = P parenleftBig X- μ σ ≤ x- μ σ parenrightBig = φ parenleftBig x- μ σ parenrightBig where φ ( . ) denotes the CDF of a standard normal. 6 5 Joint PDF Joint PDF of two continuous RV X and Y : f X,Y ( x,y ). P ( x ≤ X ≤ x + δ, y ≤ Y ≤ y + δ ) ≈ f X,Y ( x,y ) .δ 2 P ( A ) = integraldisplay integraldisplay A f X,Y ( x,y ) dxdy E [ g ( X,Y )] = integraldisplay ∞-∞ integraldisplay ∞-∞ g ( x,y...
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ReviewPart2 - 1 Probability Density Functions(PDF For a...

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