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Home Question Bank Math Probability Archive 2025 January 19

Probability Questions from Jan 19,2025

Browse the Probability 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 \( E(T) \) An urn contains 9 white balls and 8 green balls. If Tim chooses 5 balls at random from the urn, what is the probability that he will select 3 white balls and 2 green balls? Round your answer to 3 decimal places. (If necessary, consult a list of formulas.) A history class is comprised of 6 female and 5 male students. If the instructor of the class randomly chooses 7 students from the class for an oral exam, what is the probability, that 3 female students and 4 male students will be selected? Round your answer to 3 decimal places. (If necessary, consult a list of formulas.) An urn contains 9 green balls and 4 yellow balls. If Juan chooses 10 balls at random from the urn, what is the probability that he will select 7 green balls and 3 yellow balls? Round your answer to 3 decimal places. (If necessary, consult a list of formulas.) Events \( A \) and \( B \) are mutually exclusive. Suppose event \( A \) occurs with probability. 0.12 and event \( B \) occurs with probability 0.42 . Compute the following. (If necessary, consult a list of formulas.) (a) Compute the probability that \( B \) occurs or \( A \) does not occur (or both). (b) Compute the probability that either \( A \) occurs without \( B \) occurring or \( B \) occurs without \( A \) occurring. \( \square \) Events \( A \) and \( B \) are mutually exclusive. Suppose event \( A \) occurs with probability 0.06 and event \( B \) occurs with probability 0.41 . Compute the following. (If necessary, consult a list of formulas.) (a) Compute the probability that \( A \) occurs but \( B \) does not occur. (b) Compute the probability that neither the event \( A \) nor the event \( B \) occurs. \( \square \) Events \( A \) and \( B \) are mutually exclusive. Suppose event \( A \) occurs with probability, 0.43 and event \( B \) occurs with probability 0.18 . Compute the following. (If necessary, consult a list of formulas.) (a) Compute the probability that \( B \) occurs but \( A \) does not occur. (b) Compute the probability that either \( B \) occurs without \( A \) occurring or \( A \) and \( B \) both occur. \( \square \) It is estimated that \( 30 \% \) of all adults in the United States invest in stocks and that \( 85 \% \) of U.S. adults have investments in fixed income instruments (savings accounts, bonds, etc.). It is also estimated that \( 25 \% \) of U.S. adults have investments in both stocks and fixed income instruments. Answer the questions below. (If necessary, consult a list of formulas.) (a) What is the probability that a randomly chosen U.S. adult invests in stocks, given that s/he invests in fixed income instruments? Round your answer to 2 decimal places. (b) What is the probability that a randomly chosen stock investor also invests in fixed income instruments? Round your answer to 2 decimal places. \( \square \) It is estimated that \( 89 \% \) of senior citizens suffer from sleep disorders and \( 7 \% \) suffer from anxiety. Moreover, \( 6 \% \) of senior citizens suffer from both sleep disorders and anxiety. Answer the questions below. (If necessary, consult a list of formulas.) (a) Find the probability that a senior citizen suffers from anxiety, given that he or she has sleep disorder. Round your answer to 2 decimal places. (b) Given that a senior citizen suffers from anxiety, what is the probability that he or she also suffers from sleep disorder? Round your answer to 2 decimal places. It is estimated that \( 89 \% \) of senior citizens suffer from sleep disorders and \( 7 \% \) suffer from anxiety. Moreover, \( 6 \% \) of senior citizens suffer from both sleep disorders and anxiety. Answer the questions below. (If necessary, consult a list of formulas.) (a) Find the probability that a senior citizen suffers from anxiety, given that he or she has sleep disorder. Round your answer to 2 decimal places. (b) Given that a senior citizen suffers from anxiety, what is the probability that he or she also suffers from sleep disorder? Round your answer to 2 decimal places. \( \square \)
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