week4_F10

week4_F10 - Online EE131A TA session : Week 4 TA: Ni-Chun...

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Online EE131A TA session : Week 4 TA: Ni-Chun Wang nichun@ucla.edu

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Review • Probability density function (pdf) () ( ) ) X X dF x F x x F x x +Δ − ⎛⎞ ( ) ( ) XX X X fx dx x f xxFx x Fx P x X x x = ⎜⎟ Δ ⎝⎠ ⇔Δ + Δ = < + Δ – [Properties] ( ) 0 ( ( ) i d i ) F 1. () 0( is non-decreasing) 2. ( ) ( ) b a f x F x Pa X b f xdx ≤≤= 3. ( ) ( ) x ftd t −∞ = 4. ( ) 1 t −∞ = 2
• 1. A random variable X has pdf: 2 (1 ) 0 1 () 0 X cx x x fx elsewhere −≤ = (a) Find the constant c. (b) Write out the cdf F X (x). ) ind P[0<X<0 5] P[X 1] (c) Find P[0<X<0.5], P[X=1]. 1 2 0 (a) ( ) 1 ) 1 X f xdx cx x dx −∞ =⇒ = ∫∫ 1 24 0 11 1 x ⎛⎞ ⇒−= ⎜⎟ ⎝⎠ 4 c ⇒= 3

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• 1. A random variable X has pdf: 2 (1 ) 0 1 () 0 X cx x x fx elsewhere −≤ = (a) Find the constant c. (b) Write out the cdf F X (x). ) ind P[0<X<0 5] P[X 1] (c) Find P[0<X<0.5], P[X=1]. 2 4( 1 ) 0 1 (b) ( ) e l s e w h e r e X xx x = 22 4 0 0 elsewhere 1 ) 2 1 XX F x f t dt t t dt x x −∞ == = ∫∫ 24 1, 2 , 0 1 0, elsewhere X x Fx x x x > ⇒=− 4
• 1. A random variable X has pdf: 2 (1 ) 0 1 () 0 X cx x x fx elsewhere −≤ = (a) Find the constant c. (b) Write out the cdf F X (x). ) ind P[0<X<0 5] P[X 1] (c) Find P[0<X<0.5], P[X=1]. (c) P[X =1] = 0 P[0<X<0.5] = F(0.5) – F(0) – P(X=0.5) = 0.4375 - 0 - 0 = 0.4375 5

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Review : pmf, pdf, and cdf of discrete RVs • Probability Mass Function (pmf) – p (x ) = P(X=x ) X k k • cdf of discrete RVs X k k Xk xx k U(x-x ), F (x)=P(X x)= p (x ) p (x ) - < x < =∞ ∑∑ – U: step function • pdf of discrete RVs F ( ) X k XX k k (x-x ), dF (x) f( x ) = = p( x) -< x < dx δ δ : Dirac delta function 6
• 2. A modem transmits a +2 voltage signal X into a l T h hl d d t t h i i li t channel. The channel adds to this signal a noise term N that is drawn from the set {0, -1, -2, -3} with

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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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week4_F10 - Online EE131A TA session : Week 4 TA: Ni-Chun...

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