15_Exam2_review_Spring_2012

Let 1 if k th toss is h xk 0 if k th toss is t n then

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Unformatted text preview: ∑ ( x − E [ X | A ]) p X | A ( x ) ⎣ ⎦x Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x 2 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ = ∑ ( x − E [ X | Y = y ]) p X |Y ( x | y) ⎣ ⎦x ⎛ ⎞ 2 = E ⎡ X 2 | Y = y ⎤ − ( E [ X | Y = y ]) = ∑ x 2 p X |Y ( x | y) − ⎜ ∑ xp X |Y ( x | y)⎟ ⎣ ⎦ ⎝ ⎠ x 2 x E [ X | A ] = ∑ xp X | A ( x ) x 2 2 var( X | A) = E ⎡( X − E [ X | A ]) | A ⎤ = ∑ ( x − E [ X | A ]) p X | A ( x ) ⎣ ⎦x ⎛ ⎞ 2 = E ⎡ X 2 | A ⎤ − ( E [ X | A ]) = ∑ x 2 p X | A ( x ) − ⎜ ∑ xp X | A ( x )⎟ ⎣ ⎦ ⎝x ⎠ x 2 Ilya Pollak Discrete uniform random variable If X is uniform on 0,1,…, n, then Ilya Pollak Mean of a discrete uniform random variable Ilya Pollak Bernoulli PMF Toss a coin with P(H) = p, and let ⎧1 if H X =⎨ ⎩0 if T ⎧ p if k = 1 ⎪ Then pX ( k ) = ⎨1 − p if k = 0 ⎪ 0 otherwise ⎩ Ilya Pollak Mean and variance of a Bernoulli random variable Ilya Pollak Binomial PMF X = number of H's in n independent coin tosses with P(H) = p ⎛n⎞ k Then p X ( k ) = ⎜ p (1 − p )n − k , k = 0,1,…, n ⎝k⎟ ⎠ Ilya Pollak Mean and Variance of a...
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This note was uploaded on 05/28/2012 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue.

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