# 041 slides m chen iecuhk engg2430c lecture 9 7 20

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

## This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online