hw51 - y 2 dy 2 dy 1 for all-∞< l 1 ≤ u 1< ∞...

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MAE 288A Assignment 1 Due Thursday, 8 Oct., 2009 1. (5) Let f : IR 2 IR be f ( x ) = x 2 1 + x 2 2 . What is f - 1 ([2 , 4])? 2. (5) let Ω = { a, b, c, d, e, f } . Let X ( a ) = 1, X ( b ) = 1, X ( c ) = 2, X ( d ) = 2, X ( e ) = 3, X ( f ) = 3. Let Y ( a ) = 1, Y ( b ) = 3, Y ( c ) = 5, Y ( d ) = 7, Y ( e ) = 3, Y ( f ) = 5. Suppose P ( { a } ) = P ( { b } ) = P ( { c } ) = 0 . 2, P ( { d } ) = P ( { e } ) = 0 . 15, P ( { f } ) = 0 . 1. What is P ( X ∈ { 1 , 2 } , Y { 5 , 7 } )? What is P ( X ∈ { 1 , 3 } | Y ∈ { 3 } )? 3. (5) Suppose w is a scalar normal random variable with mean zero and variance, q . Consider E [exp { kw 2 / 2 } ] where k is a given positive con- stant. For what values of k is this finite? When it is finite, what is it? 4. (5) X 1 and X 2 are scalar random variables. Suppose there exists a continuous function, p : IR 2 IR such that P ( X 1 u 1 , X 2 u 2 ) = integraldisplay u 1 -∞ integraldisplay u 2 -∞ p ( y 1 , y 2 ) dy 2 dy 1 for all ( u 1 , u 2 ) IR 2 . Prove that P ( X 1 [ l 1 , u 1 ] , X 2 [ l 2 , u 2 ]) = integraldisplay u
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Unformatted text preview: , y 2 ) dy 2 dy 1 for all-∞ < l 1 ≤ u 1 < ∞ and-∞ < l 2 ≤ u 2 < ∞ . (You may use the result obtained in class if it is helpful.) 5. (5) Suppose ξ ∼ N (1 , 4). Let K . = { , 1 , 2 . . . } . Suppose { w k } k ∈K is an IID sequence of random variables with w k ∼ N (0 , 2) for all k ∈ K . Suppose ξ k +1 = 0 . 5 ξ k + w k for all k ∈ K . What is the distribution of ξ 1 ? ξ 2 ? ξ 4 ? Obtain update formulas for parameters describing the dis-tribution of ξ k +1 in terms of the parameters describing the distribution of ξ k . 1...
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