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Home Question Bank Math Archive 2025 January 19

Math Questions from Jan 19,2025

Browse the Math Q&A Archive for Jan 19,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Exercice 2 1. Soit \( N \) une variable aléatoire gaussienne centrée de variance \( \sigma^{2} \). Calculer \( E\left(e^{N}\right) \) en fonction de \( \sigma^{2} \). 2. Pour \( \gamma>0 \) et \( t>0 \), soit \( Z_{t}=e^{\gamma B_{t}-\chi^{2} t} \). Montrer que le procossus \( \left(Z_{t}\right)_{t \geqslant 0} \) est une martingale. 3. Soient \( a>0 \) et \( T=\inf \left\{t \geqslant 0: B_{t}=a\right\} \). Véridier que \( T \) est un temps d'arrôt par rapport à la filtration \( \left(F_{z}\right)_{(t>0)} \). 4. Expliquer pourquoi \( \left(Z_{t \wedge T}\right)_{t \geqslant 0} \) est une martingale, en déduire la valeur de \( E\left(Z_{t \wedge T}\right) \) pour tout \( t \geqslant 0 \). 5. Montrer qu'il existe une constante \( C>0 \), dépend uniquement de \( \gamma \) et \( a \), telle que \( Z_{t \wedge T} \leqslant C e^{\left(-\frac{\gamma^{2}(1 \wedge T)}{2}\right)} \) En déduire que \( \forall t \geqslant 0 \quad\left|Z_{i \wedge T}\right| \leqslant C \) 6. Déterminer \( \lim _{t \rightarrow \infty} Z_{i \wedge T(\omega)}(\omega) \) dans les cas suivant : (a) pour tout \( \omega \in \Omega \) tel que \( T(\omega)=\infty \). (b) pour tout \( \omega \in \Omega \) tel que \( T(\omega)<\infty \). On donne le résultat en fonction de \( Z_{T(\omega)}(\omega) \). 7. En déduire la limite (presque sûre), lorsque \( t \rightarrow \infty \), de \( Z_{t \wedge T} \), 8. Déduire des questions précédentes que \( E\left(1_{\{T<\infty\}} Z_{T}\right)=1 \). 9. Exprimer \( Z_{T} \) en fonction de \( \gamma, a \) et \( T \). 10. En prenant \( \gamma=\frac{1}{n} \) où \( n \in \mathbb{N}^{*} \). En faisant tendre \( n \rightarrow \infty \), montrer que \( P(T<\infty)=1 \). 11. Déduire des questions précédentes \( \forall \gamma>0 \quad E\left(e^{\left(-\frac{1}{2} \gamma^{2} T\right)}\right)=e^{-\gamma a} \) 12. En déduire l'expression de la transformée de Laplace de \( T: L(\alpha)=E\left(e^{-\alpha T}\right) \) 13. Déterminer \( E(T) \) Lessons 1,2,3,4 Test Due \( 03 / 23 / 25 \) 25 points possible Answered: \( 6 / 25 \) Question 7 If there are 4750 students at Rocky Valley College and \( \frac{1}{2} \) of the students take speech, how many students take speech at Rocky Valley College? Give your answer as a whole number. a) \( \sin \frac{4 \pi}{9} \) Multiply. \[ (z-10)(z+10) \] Simplify your answer. A room is \( 13 \frac{3}{4} \) feet wide and 8 feet long Find the area of the room 6. The equation \( d=(1 / 2) t \) represents the distance in meters that adult ducks walk in \( t \) seconds. a. Plot this equation in the same grid as you did the equation for the baby ducks. b. Plot the point that matches the unit rate in this situation. c. How can you see from the graphs that the adult ducks walk faster than the babies? d. How much farther will the adult ducks have walked than the baby ducks at \( t=5 \mathrm{~s} \) ? e. How much longer will the baby ducks take to walk 5 meters than the adult ducks? Multiply. \[ (z-10)(z+10) \] Describing Steps in solving a Linear Inequality Below you will find the completed steps to a problem with directions "Solve the linear inequality, show the solution on a number line, and write the solution in interval notation." Write a description of what happened in each step and answer the questions. Solve the linear inequality \( \begin{array}{ll}\text { 1. } 3(x-2)<3-2(2 x-6) & \text { a) From step one to step two: } \\ \text { 2. } 3 x-6<3-4 x+12 & \text { b) From step two to step three: } \\ \text { 3. } 3 x-6<-4 x+15 & \text { c) From step three to step four: } \\ \text { 4. } 7 x-6<15 & \text { d) From step four to step five: } \\ \text { 5. } 7 x<21 & \text { e) From step five to step six: }\end{array} \) A parade director has 48 people and needs to arrange them in rows, each row with the same number of people. Which of the following row sizes could she use? Select ALL that she could use. ? \( \square 2 \) \( \square 5 \) \( \square_{15} \) Question 4 A supermarket has a row of 30 avocados. Francis numbers those avocados from 1 through 30 and buys the first 4 whose numbers are multiples of 5 . List the numbers that correspond to these avocados (in order with commas separating them ... do not include any spaces).
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