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09_Midterm_sol

09_Midterm_sol - EECE453 Midterm 2009 Question 1 1 If A and...

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Unformatted text preview: EECE453 Midterm 2009 Question 1 1. If A and B are independent events, prove that A and B g are also independent (here B g denote the complement of event B) [sol] G¡¢, £ ¤ ¥ = G[¢,¡¦ − £¥] , where C = whole set = G¡¢¥ − G¡¢,£¥ = G¡¢¥ ∙ [1 − G¡£¥] = G¡¢¥G¡£ ¤ ¥ therefore, A and B g are also independent. 2. A binary source § ∈ [0,1] transmits with probabilities P(X=0) =3/4 through a channel. Due to noise, the output Y differs from X occasionally. The channel output Y is modeled by conditional probabilities: P(Y=1|X=1) = p 1 and P(Y=0|X=0) = p 1 . where p 1 is the parameter you have computed on page 1. Compute P(Y=1), P(Y=0), and P(X=1|Y=1). [sol] channel matrix ¦ = ¨ © ª 1 − © ª 1 − © ª © ª « = ¨ 2/14 12/14 12/14 2/14 « ¬ = ¨ 3/4 1/4 «,­ = ® ¯ − ¬ = ¨ 9/28 19/28 « = ¨ © ª 1 − © ª 1 − © ª © ª «¨ 3/4 1/4 « G¡­ = 1¥ = ª° ±² ,G¡­ = 0¥ = ° ±² G¡§ = 1|­ = 1¥ = ³¡´µª,¶µª¥ ³¡¶µª¥ = · ¸ ¹ º ·¸ ·» º¼ = ª ª° 3. A binary source X[K] has transition probability P{(X[K])(X[K-1])} gives by the following matrix 3....
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