Assignment-daixieTM为您提供佛罗里达大学University of Florida STA 4322 Mathematical Statistics 2数学统计代写代考和辅导服务！
Instructions:
Mathematical statistics is a branch of mathematics that deals with the theoretical foundation of statistical methods and their applications. It involves the development of mathematical models, algorithms, and techniques for analyzing and interpreting data.
Mathematical statistics is concerned with understanding the behavior of statistical procedures and their properties, such as consistency, efficiency, and unbiasedness. It involves the development of probability theory, including the study of probability distributions, hypothesis testing, estimation, and regression analysis.
Mathematical statistics is used in a wide range of fields, including engineering, physics, finance, biology, and social sciences. It provides tools for understanding and modeling complex systems and making decisions based on data. Some of the key techniques used in mathematical statistics include Bayesian analysis, maximum likelihood estimation, and Monte Carlo methods.
Two sisters maintain that they can communicate telepathically. To test this assertion, you place the sisters in separate rooms and show sister A a series of cards. Each card is equally likely to depict either a circle or a star or a square. For each card presented to sister A, sister B writes down ‘circle’, or ‘star’ or ‘square’, depending on what she believes sister A to be looking at. If ten cards are shown, what is the probability that sister B correctly matches at least one?
We will calculate a probability under the assumption that the sisters are guessing. The probability of at least one correct match must be equal to one minus the probability of no correct matches. Let $F_i$ be the event that the sisters fail to match for the ith card shown. The probability of no correct matches is $\mathrm{P}\left(F_1 \cap F_2 \cap \ldots \cap F_{10}\right)$, where $\mathrm{P}\left(F_i\right)=\frac{2}{3}$ for each $i$. If we assume that successive attempts at matching cards are independent, we can multiply together the probabilities for these independent events, and so obtain
$$
\mathrm{P}\left(F_1 \cap F_2 \cap \ldots \cap F_{10}\right)=\mathrm{P}\left(F_1\right) \mathrm{P}\left(F_2\right) \ldots \mathrm{P}\left(F_{10}\right)=\left(\frac{2}{3}\right)^{10}=0.0173 .
$$
Hence the probability of at least one match is $1-0.0173=0.9827$.
Let $A$ be the event that you give the correct answer. Let $B$ be the event that you knew the answer. We want to find $\mathrm{P}(A)$. But $\mathrm{P}(A)=\mathrm{P}(A \cap B)+\mathrm{P}\left(A \cap B^c\right)$ where $\mathrm{P}(A \cap B)=\mathrm{P}(A \mid B) \mathrm{P}(B)=1 \times 0.75=0.75$ and $\mathrm{P}\left(A \cap B^c\right)=\mathrm{P}\left(A \mid B^c\right) \mathrm{P}\left(B^c\right)=\frac{1}{5} \times 0.25=0.05$. Hence $\mathrm{P}(A)=0.75+0.05=0.8$.
Three babies are given a weekly health check at a clinic, and then returned randomly to their mothers. What is the probability that at least one baby goes to the right mother?
Let $E_i$ be the event that baby $i$ is reunited with its mother. We need $\mathrm{P}\left(E_1 \cup E_2 \cup E_3\right)$, where we can use the result
$$
\operatorname{Pr}(A \cup B \cup C)=\operatorname{Pr}(A)+\operatorname{Pr}(B)+\operatorname{Pr}(C)-\operatorname{Pr}(A \cap B)-\operatorname{Pr}(A \cap C)-\operatorname{Pr}(B \cap C)+\operatorname{Pr}(A \cap B \cap C) .
$$
for any $\mathrm{A}, \mathrm{B}, \mathrm{C}$. The individual probabilities are $\mathrm{P}\left(E_1\right)=\mathrm{P}\left(E_2\right)=\mathrm{P}\left(E_3\right)=\frac{1}{3}$. The pairwise joint probabilities are equal to $\frac{1}{6}$, since $\mathrm{P}\left(E_1 E_2\right)=\mathrm{P}\left(E_2 \mid E_1\right) \mathrm{P}\left(E_1\right)=\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)$, and the triplet $\mathrm{P}\left(E_1 E_2 E_3\right)=\frac{1}{6}$ similarly. Hence our final answer is
$$
\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{6}-\frac{1}{6}-\frac{1}{6}+\frac{1}{6}=\frac{2}{3}
$$
