041 slides m chen iecuhk engg2430c lecture 9 7 20

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Unformatted text preview: )=y for every possible value of y (revised from MIT opencourse 6.041 slides.) M. Chen ([email protected]) ENGG2430C lecture 9 7 / 20 Derived Distribution: Example Let X be a discrete random variable with PMF pX (x ) = |x |/a, if x = −2, −1, 0, 1, 2; 0, otherwise. Let Y = 2X + 1. Find the PMF of Y and plot it. M. Chen ([email protected]) ENGG2430C lecture 9 8 / 20 Derived Distribution: Continuous R.V. Cases Consider a continuous r.v. X and Y = g (X ): its CDF is given by ˆ fX (x ) dx FY (y ) = P (Y ≤ y ) = x :g (x )≤y Consequently, its PDF is given by fY (y ) = dFY (y ) dy Example: X ∼ U [−1, 1], find PDF of Y = X 2 : √ √ √1 FY (y ) = P (X 2 ≤ y ) = P (− y ≤ X ≤ y ) = 2 y · ∀y ∈ [0, 1] 2 1 fY (y ) = √ , ∀y ∈ (0, 1] 2y M. Chen ([email protected]) ENGG2430C lecture 9 9 / 20 Derived Distribution: Example * X and Y are two random variables with PDF: (From MIT opencourse 6.041 slides.) Let Z = Y /X (it is a r.v.), computes its PDF. (Hint: we compute CDF of Z and then its PDF. P (Z ≤ z ) = P (Y ≤ z · X ). Compute P (Y ≤ z · X ) for two cases separately: z ∈ (0, 1) and z ∈ [1, ∞).) M. Chen ([email protected]) ENGG2430C lecture 9 10 / 20 Derived Distribution: Max/Min of R.V.s Consider two independent r.v.s X and Y , let Z = max(X , Y )...
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