MECS Ex1Sol - : : :1 . 10 ( ) .1 . 5 ( . ( : ( P( X = x) =...

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Unformatted text preview: : : :1 . 10 ( ) .1 . 5 ( . ( : ( P( X = x) = f ( x; ) = 1 e -x e 10 = 10 -x P( X 5) = f ( x; ) dx = 5 5 e = -e 10 10 -x -x 10 = e - 0.5 = 0.3935 5 ( P( X = x) = f ( x; ) = 1 e -x = P( X x0 ) = f ( x; ) dx = x0 1 x0 e -x = -e -x xo =e - x0 1 : : :1 . P( X 1 + X 2 > 2) = 1 X 2 X 1 .2 : e t0 P( X 1 = t ) = P( X 2 = t ) = f (t ) = 0 else -t : Z = X1 + X 2 : P ( Z = z ) = P( X 1 = t X 2 = z - t ) = f Z ( z ) = f (t ). f ( z - t ) dt - + = e e 0 z -t - z + t dt =z.e + -z : P ( Z > 2) = g Z ( z )dz = - ( z + 1)e 2 - z + z =2 = 3.e - 2 = 0.406 2 : : :1 . 1 / X 2 X 1 .3 : f X 1 + X 2 (t ) = 2 te - t , t 0 (. 2 X 1 + X 2 ) 1 / X 1 , X 2 ,..., X n : n X 1 , X 2 ,..., X n +1 f X 1 + X 2 +...+ X n+1 (t ) = 2 te - t (t ) n / n!, t 0 : e - t f X 1 (t ) = f X 2 (t ) = 0 Z = X1 + X 2 if t 0 else f X 1 + X 2 ( z ) = P( Z = z ) = P[( X 1 = t ) ( X 2 = z - t )] = - z f X 1 (t ) f X 2 ( z - t )dt = (e - t )(e - ( z -t ) )dt 0 z = 2 e - z d t = 2 e - z z , 0 z0 f X 1 + X 2 (t ) = 2 e - t t , t0 : k n . k n : Z n +1 = X 1 + X 2 + ... + X n + X n +1 : f X1+ X 2 +...+ X n + X n+1 ( z ) = P( Z n+1 = z ) = P[( Z n = t ) ( X n+1 = z - t )] = f X1+ X 2 +...+ X n (t ) f X n+1 ( z - t )dt = ( - 0 z 2te -t (t ) n-1 (n - 1)! = n! )(e - ( z -t ) )dt zn = 2 .n-1.e -z (n - 1)! z 0 t n-1dt = 2 .n-1.e -z t (n - 1)! . n t=z 2 .n-1.e -z n t =0 = 2te -t (t ) n / n! 3 : : :1 X 1 1 / 2 1 / 1 X 2 X 1 .4 : ( ) X 2 P{ X 1 < X 2} = 1 + 2 : 1 e t0 f X 1 (t ) = 1 0 else e - 2t t 0 f X 2 (t ) = 2 0 else : - 1t Z = X 2 - X1 : P ( X 1 < X 2 ) = P ( Z > 0) : P( Z = z ) = f Z ( z ) = P( X 2 = t X 1 = t - z ) = f X 1 ( z + t ) f X 2 (t ) dt = 1e -1 ( t - z ) .2 e -2 .t dt - -z + - ( 1 + 2 ).t t = t=z = - 1.2 e 1 . z = 2 e 1 + 2 = 1.2 e 1 . z . e -( 1 + 2 ). z 1 + 2 1.2 - . z e 1 + 2 . f Z ( z ) = 1 2 e - . z 1 + 2 2 P ( Z > 0) = f Z ( z ) dz = 0 0 1.2 - . z e dz 1 + 2 2 = 1 + 2 1 z = e - 2 . z z =0 = 1 + 2 1 4 : : :1 . .5 : P(T > r + s | T > r ) = P(T > s ) : g ( t ; p ) = p .( 1 - p ) P (T > r + s ) = t =r + s t g (t , p) = (1 - p) p t =r + s p.(1 - p) t = p (1 - p ) t = p t =r + s r+s = (1 - p ) r + s (5 - 1) : (5 - 2) P(T > r + s ) = (1 - p) r + s P (T > r ) = (1 - p) r P (T > s ) = (1 - p) P (T > r + s | T > r ) = P (T > r + s | T > r ) = P(T > s ) s (5 - 3) : (5 - 3) (5 - 2) (5 - 1) P(T > r + s ) (1 - p ) r + s = = (1 - p ) s = P(T > s ) P(T > r ) (1 - p ) r 5 : : :1 . 1.2 .6 ( ( : ( p ( k ; t ) = ( t ) k e - ( t ) k! (1.2) 0 e -1.2 = 0.301 p (0;1.2) = (0)! ( p (0;2.4) = ( 2 .4 ) e (0)! 0 -2.4 = 0.091 6 : : :1 : 0.8 .7 ( ( : ( e - n n! P( X > 2) = 1 - P( X = 0) - P( X = 1) - P( X = 2) P ( X = n) = p(n, ) = = 1 - p(0, ) + p(1, ) + p(2, ) e -0.8 (0.8) 0 e -0.8 (0.8)1 e -0.8 (0.8) 2 = 1- - - 0! 1! 2! = 0.047 ( P ( X = 1) = p(1;0.8) = e -0.8 (0.8)1 = 0.359 1! 7 : : :1 0.4 0.6 10 .8 : 50 10 ( 50 100 10 ( : Y X Z = X + Y P ( X = n) = p (n; pt ) = e -6t (6t ) n n! e - 4t (4t ) n P (Y = n) = p (n; (1 - p)t ) = n! ( P( X = 50) = p(50;40) = e -40 4050 = 0.018 50! . () ( 8 : : :1 . = 1 1 - p p .9 : 9 : : :1 . = 1 + 2 2 1 .10 : e - 1 1 x0 P ( X 1 = x ) = f X 1 ( x ) = x! 0 else x e - 2 2 x x0 P ( X 2 = x ) = f X 2 ( x ) = x! 0 else Z = X +Y e - 1 1 e - 2 2 P ( Z = z ) = f Z ( z ) f X 1 ( x ). f X 2 ( z - x ) = . x! ( z - x )! x = - x =0 + z x z-x = e - 1 e - 2 e z x =0 z 1 x x! ( z - x )! . 2 - x = e - 1 e - 2 e z x =0 z 1 x x! ( z - x )! : . 2 - x = e - 1 e - 2 z z! 1 1 x! ( z - x )! x =0 2 z x e - 1 e - 2 z = z! 1 e - ( 1+ 2 ) (1 + 2 ) z 1 + = z! 2 z e - ( 1+ 2 ) (1 + 2 ) x f Z ( x) = x! . 1 + 2 10 ...
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