zz-12 - j + 1)st toss is diFerent from the outcome of the j...

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Utah State University ECE 6010 Stochastic Processes Homework # 3 Due Friday Sept. 23, 2005 1. Suppose X and Y are the indicator functions of events A and B , respectively. Find ρ ( X, Y ), and show that X and Y are independent if and only if ρ ( X, Y ) = 0. 2. Suppose φ ( u ) is a ch.f. Show that | φ ( u ) | 2 is also a ch.f. 3. Suppose X and Y are jointly Gaussian. Use ch.f.s to show that ρ ( X, Y ) = ρ . 4. Suppose X and Y are jointly continuous. (a) Show that F Y | X ( b | x ) = Z b -∞ f XY ( x, y ) f X ( x ) dy and thus that f Y | X ( y | x ) = f XY ( x, y ) f X ( x ) (b) Suppose R -∞ | y | f Y | X ( y | x ) dy < . Show that E [ Y | X = x ] = R -∞ yf Y | X ( y | x ) dy. 5. Suppose X and Y are independent continuous r.v.s with c.d.f.s F X and F Y , respectively. Suppose further that F X ( b ) F Y ( b ) for all b R . Show that P ( X Y ) 1 / 2 . 6. Prove Jensen’s inequality for the case of simple-function r.v.’s 7. Prove the Schwartz inequality. Problems from : 1. Prob 2.7.4 2. Prob 2.7.7. Hint: binomial distribution 3. Prob 2.7.9. Hint: P ( X - x ) = ( 0 x < 0 , 1 - lim y ↑- x F ( y ) x 0 . 1
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4. Ex 3.3.1. Hint: Let p X ( - 1) = 1 9 , p X ( 1 2 ) = 4 9 and p X (2) = 4 9 . 5. Ex 3.4.1. Let I j be the indicator function of the event that the outcome of the (
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Unformatted text preview: j + 1)st toss is diFerent from the outcome of the j th toss. The number R of distinct runs is R = 1 + n-1 j =1 I j . Observe that I j and I k are independent if | j-k | &gt; 1. Show that E [( R-1) 2 ] = ( n-1) E [ I 1 ] + 2( n-2) E [ I 1 I 2 ] + (( n-1) 2-( n-1)-2( n-2)) E [ I 1 ] 2 . Show that E [ I 1 I 2 ] = p 2 q + pq 2 = pq. 6. Ex. 3.4.2. Hint: Let T = k i =1 X i , where X i is the number on the i th ball. Show that: E [ T ] = 1 2 k ( n + 1). show that E [ T 2 ] = 1 6 k ( n + 1)(2 n + 1) + 1 12 k ( k-1)(3 n + 2)( n + 1). Hint: N X k =1 k = n ( n + 1) 2 n X k =1 k 2 = n ( n + 1)(2 n + 1) 6 . n X k =1 k 3 = n ( n + 1) 2 2 . 7. Ex 3.5.2. Hint: P ( H = x ) = n = x P ( H = x | N = n ) P ( N = n ). 8. Ex 3.6.5. Hint: log y y-1, with equality if and only if y = 1. 2...
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zz-12 - j + 1)st toss is diFerent from the outcome of the j...

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