Assignment-daixieTM为您提供伯明翰大学University of Birmingham Mathematical Methods for Economics 07 33189经济学的数学方法代写代考和辅导服务！
Instructions:
Advanced mathematical concepts related to economics and econometrics! You will be building on the mathematical foundation that you have gained in previous courses and exploring the relationship between mathematics and economics more deeply.
In unconstrained optimization, you will be examining functions with many variables, starting with quadratic functions and then gradually generalizing both the results and methods. You will also be looking at how these methods can be applied to econometric estimation.
Under constrained optimization, you will be considering several important topics in economics, such as problems with equality and inequality constraints, dynamic optimization (which involves time), and methods of resource allocation, such as Linear Programming. You will also be exploring other topics of importance to economics and econometrics, such as comparative statics, convexity, and the Envelope Theorem.
Overall, it sounds like you will be gaining a deeper understanding of how mathematics can be applied to economic problems, and how optimization is a dominant theme in both economics and econometrics. Good luck with your studies!
Don’t spend much time on these questions, a short answer to each suffices. (a) For a sequence of $n$ independent trials, each of which can result in a “success” (with probability $p$ ) or a “failure” (probability $1-p$ ), what is the p.d.f. $f_X(x)$ of the total number $x$ of successes? Be careful about specifying the p.d.f. for all real numbers.
The total number of successes $X$ follows a binomial distribution with parameters $n$ and $p$, so the p.d.f. of $X$ is:
$f_X(x)=\left(\begin{array}{l}n \ x\end{array}\right) p^x(1-p)^{n-x} \quad$ for $x=0,1, \ldots, n$.
(b) You are given the joint p.d.f. $$ f_{X Y}(x, y)=\left\{\begin{array}{cl} c \exp \{-(x+y)(x-y)\} & x \in[-2,2], y \in[-2,2] \\ 0 & \text { otherwise } \end{array}\right. $$ where $c$ is a positive constant such that the density integrates to 1 . Are $X$ and $Y$ independent?
(b) Yes, $X$ and $Y$ are independent. This can be shown by calculating the marginal p.d.f.s of $X$ and $Y$ and verifying that their product is equal to the joint p.d.f.
(c) A pregnant woman goes to her obstetrician complaining of visual disturbances. “Don’t worry”, her OB tells her, “only a small fraction of miscarriages or other adverse events are preceded by visual disturbances.” Explain briefly, preferably using a formula, why the patient should be annoyed, not reassured.
(c) The patient should be annoyed because the OB’s statement does not provide any useful information about the probability of a miscarriage or adverse event given the presence of visual disturbances. In probability notation, the OB’s statement is equivalent to $P(\text{visual disturbances} \mid \text{miscarriage}) \ll 1$, but the patient is interested in $P(\text{miscarriage} \mid \text{visual disturbances})$, which may be much larger. This is an example of the base rate fallacy, where the prior probability of an event is ignored or given insufficient weight in favor of specific diagnostic information.
