hw0-solution

hw0-solution - 3.1 a) This is an instance of the Bernoulli...

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3.1 a) This is an instance of the Bernoulli experiment. 73 7 10 1 1 120 0.117 2 2 1024 C ⎛⎞⎛⎞ == ⎜⎟⎜⎟ ⎝⎠⎝⎠ b) P(both tosses are tails | one of the tosses in a tail) = P(both tosses are tails)/P(one of the tosses in a tail) = 1/3 c) () 0 . 4 ,() 0 . 2 5 PA PB max ( ) 0.25 PA B ∩= min ( ) 0 = max ( ) 0.65 ∪= min ( ) 0.4 d) Probability density function 2 2 1( ) () e x p 2 2 x fx µ σ σπ ⎛⎞ =− ⎜⎟ ⎝⎠ mean is , variance is 2 . 3.2 a) 112314 1 1 T ab =×+×+× = b) The dot product of two vectors u and x is defined as cos T ux u x θ = where u , x are the length of vector u and x , and is the measure of the angle between vectors u and x . when 1, cos 1 u , max( ) cos T ux u x x when 1 u , min( ) cos T x = 0 u , min(| |) | cos | 0 T = = c) 2 (1 ) x x df x e dx e + b) 2 (, ) 1 2 xy fxy x ye yy =+ ⋅ c) cos( ) ( 1) sin( ) cos( ) sin( ) ff x f y yu v x u v vx vy v v x u v ∂∂ =⋅+⋅= + − − + , then plug in x and y. d) Suppose 2 x x f xa eb e =+ 2 2 x x df ae be dx , let df dx be zero, we have 2 20
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hw0-solution - 3.1 a) This is an instance of the Bernoulli...

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