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# Untitled

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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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