HW5 Solutions.pdf - EE 131A Probability and Statistics...

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EE 131A Homework 5 Solutions Probability and Statistics Monday, November 5, 2018 Instructor: Lara Dolecek Due: Monday, November 12, 2018 by 5:00 pm via CCLE TA: Siyi Yang Reading: Chapter 4 1. Solutions: The characteristic function of Y = aX + b is given by, Φ Y ( ω ) = E [ e jωY ] = E [ e ( aX + b ) ] = e jωb E [ e jaωX ] = e jωb e jamω - a 2 σ 2 ω 2 / 2 which is the characteristic function of Gaussian random variable with mean of am + b and variance of a 2 σ 2 . 2.
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Applying the Chebyshev inequality to the event {| Y - p | > a } , we have: P {| Y - p | > a } ≤ σ 2 a 2 = p (1 - p ) na 2 . As n → ∞ , P {| Y - p | > a } → 0 for any fix a > 0. 4. Solution: Since

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