t109soln - STAT 330 TEST 1 SOLUTIONS 1(a) [3] Suppose X v...

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STAT 330 TEST 1 SOLUTIONS 1( a )[3] Suppose X v POI ( θ ) . Show E ( X ( k ) )= E [ X ( X 1) ··· ( X k +1)] = θ k , k =1 , 2 ,... . E ³ X ( k ) ´ = X x = k x ( k ) θ x e θ x ! = e θ X x = k θ x ( x k )! let y = x k = e θ X y =0 θ y + k y ! = e θ θ k X y =0 θ y y ! = e θ θ k e θ by the Exponential Series = θ k ( b )[2 ] Use ( a ) to f nd Var ( X ) . ( X E ³ X (2) ´ + E ( X ) [ E ( X )] 2 = θ 2 + θ θ 2 = θ ( c )[3 ] Find M ( t E ( e tX ) , the m.g.f. of X. For what values of t does M ( t ) exist? M ( t X x =0 e tx θ x e θ x ! = e θ X x =0 ( θ e t ) x x ! = e θ e θ e t by the Exponential Series = e θ ( e t 1 ) for t ( −∞ , ) 1
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2 . Suppose X is a random variable with p.d.f. f ( x )= e x/ θ θ ¡ 1+ e x/ θ ¢ 2 , −∞ <x< , θ > 0 . ( a )[3 ] Find the c.d.f. of X . F ( x Z x −∞ e u/ θ θ ¡ e u/ θ ¢ 2 du =l i m a →−∞ 1 e u/ θ | x a ¸ i m a →−∞ 1 e x/ θ + 1 e a/ θ ¸ =1 1 e x/ θ = e x/ θ e x/ θ , ( b )[2 ] Show that θ is a scale parameter for this distribution.
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t109soln - STAT 330 TEST 1 SOLUTIONS 1(a) [3] Suppose X v...

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