week5_F10 - Online EE131A TA session : Week 5 TA: Ni-Chun...

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Online EE131A TA session : Week 5 TA: Ni-Chun Wang [email protected]
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Midterm Review (Problem sets of 2009 Summer) 2
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(a) (A man sit on the first seat) => 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1 (A woman sit on the first seat) => 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1 Total = 2 x (4 x 4 x 3 x 3 x 2 x 2 x 1 x 1) (b) Choose 1 man out of 4 to sit in the first seat, and let others choose their seat one at a time 7 4 x 7! 47 17 CP =⋅ 3
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(a) Tossing two fair coins , S: (HH),(HT),(TH),(TT) P(B1) = P(HH) = 1/4; P(B2) = P(TT) = 1/4; P(B3) = 1/2 (b) P(A|B1) = 5/10; P(A|B2) = 10/15; P(A|B3) = 5/15 (c) By theorem on total probability: P(A) = P(B1)P(A|B1) + P(B2)P(A|B2) + P(B3)P(A|B3) 1/4 x 5/10 + 1/4 x10/15 + 1/2 x 5/15 11/24 4 = 1/4 x 5/10 + 1/4 x10/15 + 1/2 x 5/15 = 11/24 (d) By Bayes’ Rule P(B1|A) = P(B1)P(A|B1)/P(A) = 1/4 x 5/10 / (11/24)
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) 8 2 ( 1 ) Let Y~N(1,0.25) (a) 0.6 0.7 0.8 1( () e x p ( ) 20 .25 2 (0.25) y gy π = × 1 ,( ) 1 d yg y dy r ∞∞ = =< 0.4 0.5 0 −∞ ∫∫ 1 Let f x g x = 0.2 0.3 0 1 1 X X fg r f x dx g x dx r ⇒= = 5 -2 -1 0 1 2 3 4 0 0.1 00
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) (a) 2 1 () exp ( ) ( ) , ~N (0 ,1 ) 2 2 t t td t P T t T π −∞ Φ= = To use the Table of function, we need to normalize Y to N(0,1), Φ 10 1 i.e. ( 0) ( ) ( -2) ( 2) 0.25 0.25 0) 1 ( 2) 0.9772 Y PY P PT Y ≤= = = Φ > =−Φ −= ( ⇒> Φ 2 So the pdf of X can be written as 11 ( 1 ) p( ) x −− 6 exp( 0.9772 2 0.25 , 0 2 (0.25) 0, X fx x elsewhere × =≥
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) (b) 1.1 0.9 (0. 91 .1) ( 1.1) ( 0.9) 1 PX ≤≤ =≤ −< ⎛⎞ 00 () 0.9772 11 . 1 1 0 . 9 1 ( ) gx d xg x dx =− ⎜⎟ ⎝⎠ −− Φ ∫∫ ) 1 ( ) x = Φ 0.977 20 .5 0.5 (0.2) ( 0.2) 1 ( 0.2) ( 0.2) 0.1617 0.9772 0.9772 = Φ −Φ − = −Φ − −Φ − = xx Φ − = −Φ 7
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) ( ) x x f t t 5 0 5 (a )1 1 1 ( x x X ce dx c xdx e f −∞ ⇒= ⎛⎞ ⇒− = ⎜⎟ = 5 0 () (c) ( ) )5 x t X X Fx ed dt t −∞ = = 0 5 5 c ⎝⎠ (b) ( ) b Pa X b x f dx ≤≤= ( ) 55 0 1, 0 t x x e ex =≥ = 5 0 ) x x −≥ 2 (1 2) a X PX fx d x << = 0, X elsewhere = 1 2 1 0 1 x ee e −− =− = 8
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) i i l (a) : () ( ) ( 1 - ) , 0 , 1 ,2 nk n k Xk Binomial pk P X k C p p k = == = " 202 0 211 1 20 (0) (0.6) (0.4) 0.16 (1) (0.6) (0.4) 0.48 pC 220 2 (2) (0.6) (0.
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This note was uploaded on 11/05/2010 for the course ELECTRICAL EE131A taught by Professor Kungyao during the Spring '10 term at UCLA.

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week5_F10 - Online EE131A TA session : Week 5 TA: Ni-Chun...

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