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  1. \begin{document}
  2.  
  3. \maketitle
  4.  
  5. \section{Calcul de lois}
  6.  
  7. Question 1 :
  8. \newline
  9. \\
  10. X a pour densité sur \mathbb{R} , f\textsubscript{X}(x) = \lambda e \up{$-(\lambda x$ + e\up{- $$\lambda$ x})}
  11. \\
  12. \Phi (x) = 3x+2
  13. \\
  14. Z = \Phi (x)
  15. \newline
  16. \\
  17. Pour ~ trouver ~la ~ densité ~ de ~ Z ~ on ~ utilise ~ la ~ fonction ~ de ~
  18. répartition
  19. \newline
  20. \\
  21. F\textsubscript{Z}(x) = P(Z \leq t)
  22. \newline
  23. F\textsubscript{Z}(x) = P(\Phi (x) \leq t)
  24. \newline
  25. F\textsubscript{Z}(x) = P(3X+2 \leq t)
  26. \newline
  27. F\textsubscript{Z}(x) = P(X \leq \frac{t-2}{3})
  28. \newline
  29. F\textsubscript{Z}(x) = $\displaystyle \int_{- \infty}^{t} \lambda e\up{$-(\lambda$ ( $\frac{x-2}{3}$) +e\up{$-\lambda$ ($\frac{x-2}{3}$)}}   \mathrm{d}x$
  30. \newline
  31. F\textsubscript{Z}(x) = \lambda $\displaystyle \int_{- \infty}^{t}  e\up{$-\lambda$ ( $\frac{x-2}{3}$) . e\up{$-\lambda$ ($\frac{x-2}{3}$)}}   \mathrm{d}x$
  32. \newline
  33. F\textsubscript{Z}(x) = 3 $\displaystyle \int_{- \infty}^{t} $\frac{\lambda}{3}$ e\up{$-\lambda$ ( $\frac{x-2}{3}$) . e\up{$-\lambda$ ($\frac{x-2}{3}$)}}   \mathrm{d}x$
  34. \newline
  35. F\textsubscript{Z}(x) = 3 [e\up{-e\up{$-\lambda(\frac{x-2}{3})$}}]_- \infty^t
  36. \newline
  37. \\
  38. F\textsubscript{Z}(x) = 3 (e\up{-e\up{$-\lambda(\frac{t-2}{3})$}})
  39. \newline
  40. \\
  41. F\textsubscript{Z}(x) = 3 (e\up{-e\up{$(\frac{14-7t}{3})$}})
  42. \newline
  43. \\
  44. f\textsubscript{Z}(x) = F\textsubscript{Z}'(x)
  45. \newline
  46. \\
  47. f\textsubscript{Z}(x) = \frac{7}{3}.e\up{$(\frac{14-7t}{3})$}} . 3(e\up{-e\up{$(\frac{14-7t}{3})$}})
  48. \newline
  49. \\
  50. f\textsubscript{Z}(x)= 7e\up{$(\frac{14-7t}{3}$-e\up{$(\frac{14-7t}{3})$}})  \\
  51.  \newline
  52.   \newline
  53.    \newline
  54.     \newline
  55.      \newline
  56.       \newline
  57.         \newline
  58.  
  59.  Question 2 :
  60.  \newline
  61. \\
  62. F\textsubscript{Z}(t) = 3 (e\up{-e\up{$-\lambda(\frac{t-2}{3})$}})
  63. \newline
  64. \\
  65. f\textsubscript{Z}(t) = F\textsubscript{Z}'(t) = \lambda e\up{$-\lambda$ ( $\frac{t-2}{3}$) .e\up{-e\up{$-\lambda(\frac{t-2}{3})$}}}
  66. \newline
  67. \\
  68. Donc $\displaystyle \int_{- \infty}^{t} f\textsubscript{Z}(x) ~ \mathrm{d}x$ = 0,5
  69. \newline
  70. \\
  71. \Leftrightarrow  3 (e\up{-e\up{$-\lambda(\frac{t-2}{3})$}}) = 0,5
  72. \newline
  73. \\
  74. \Leftrightarrow -e\up{$-\lambda(\frac{t-2}{3})$}} = ln(\frac{1}{6})
  75. \newline
  76. \\
  77. \Leftrightarrow -\lambda (\frac{t-2}{3}) = ln(-ln(\frac{1}{6}))
  78. \newline
  79. \\
  80. \Leftrightarrow - \frac{\lambda t}{3} + \frac{2 \lambda}{3} = ln(ln(6))
  81. \newline
  82. \\
  83. \Leftrightarrow - \frac{7}{3} t = ln(ln(6)) - \frac{14}{3}
  84. \newline
  85. \\
  86. \Leftrightarrow t = - \frac{3}{7} ln(ln(6)) + 2
  87. \newline
  88. \\
  89. \Leftrightarrow t \simeq 1,750
  90. \newline
  91.  
  92. Question 3 :
  93.  \newline
  94. \\
  95. X \texttildelow U([0,\lambda])
  96.  
  97. \section{Calcul d'Esperance}
  98.  
  99. Question 1 :
  100. \newline
  101. \\
  102. X \Rightarrow G(p) ~ avec ~ p $ \in $ ]0;1[
  103. \newline
  104. \\
  105. $ P(X=k) = p(1-p)\up{k-1} $
  106. \newline
  107. \\
  108.  E[e\up{-tx}] = \sum_{k=1}^\infty e\up{-tk} ~ . ~ p(1-p)\up{k-1}
  109. \newline
  110. \\
  111.  E[e\up{-tx}] = \frac{p}{1-p} \sum_{k=1}^\infty e\up{-tk} ~ . ~ p(1-p)\up{k}
  112. \newline
  113. \\
  114.  E[e\up{-tx}] = \frac{p}{1-p} \sum_{k=0}^\infty (e\up{-t} (1-p))\up{k+1}
  115. \newline
  116. \\
  117.  E[e\up{-tx}] = \frac{p}{1-p} e\up{-t} (1-p) \sum_{k=0}^\infty (e\up{-t} (1-p))\up{k}
  118. \newline
  119. \\
  120.  E[e\up{-tx}] = pe\up{-t} . \frac{1}{1-e\up{-t}(1-p)}
  121. \newline
  122. \\
  123.  E[e\up{-tx}] = \frac{p}{e\up{t}-(1-p)}
  124.  \\
  125.  
  126.  
  127.  Question 2 :
  128.  \newline
  129. \\
  130. E[e\up{-t(x+y)}] = E[e\up{-tx}.e\up{-ty}]
  131.  \newline
  132. \\
  133. E[e\up{-t(x+y)}] = E[e\up{-tx}].E[e\up{-ty}] ~ ~ car X et Y sont independants
  134. \end{document}
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