Avalie a afirmação abaixo, com relação à escolha sob incerteza:

Item 4 - Se c₁ representa o consumo no estado 1 e c₂ o consumo no estado 2, e da mesma forma p₁ representa a probabilidade do estado 1 e p₂ a probabilidade do estado 2, uma função de utilidade Von Neumann- Morgenstern assumiria a forma: !$ c_1^{p_1} C_2^{p_2} !$

Considere as matrizes

!$ A = { \begin{bmatrix}1\,\,\,\,\,\,a\\2\,\,\,-1 \end{bmatrix}}, B= { \begin{bmatrix}1\,\,\,\,\,\,b\\b\,\,\,\,1 \end{bmatrix}} !$ e !$ C = { \begin{bmatrix}cos\,\theta\,\,\,\,\,\,sen\,\theta\\-sen\,\theta\,\,\,cos\,\theta \end{bmatrix}} !$. Julgue a afirmativa:

Item 4 - C é invertível não simétrica.

When Edwin Hubble, the astronomer after whom the space telescope is named, looked at the skies in the 1920s using what was then the world’s largest telescope, he found that there were galaxies outside the Milky Way, an observation that profoundly shocked his contemporaries. Max Planck founded quantum theory, and thus wrecked the Newtonian universe. And William Herschel discovered Uranus, the first planet unknown to the astrologers of old. If the new and refurbished instruments named after these scientific mould-breakers do as much in the next few years as their eponyms did, it will, indeed, be an exciting time for astronomy and physics.

The paragraph above tells us about

Item 1 - the discovery of the Milky Way;

When Edwin Hubble, the astronomer after whom the space telescope is named, looked at the skies in the 1920s using what was then the world’s largest telescope, he found that there were galaxies outside the Milky Way, an observation that profoundly shocked his contemporaries. Max Planck founded quantum theory, and thus wrecked the Newtonian universe. And William Herschel discovered Uranus, the first planet unknown to the astrologers of old. If the new and refurbished instruments named after these scientific mould-breakers do as much in the next few years as their eponyms did, it will, indeed, be an exciting time for astronomy and physics.

The paragraph above tells us about

Item 0 - several outstanding scientists and their work;

Sobre a Teoria das Probabilidades e considerando A, B e C três eventos quaisquer, mas com probabilidades de ocorrência diferentes de zero, indique a alternativa correta e falsa:

Item 4- !$ P(A \cup B \cup C ) \le P(A) +P(B) + P(C) !$ , com desigualdade estrita se, e somente se, os eventos forem independentes.

Considere uma economia com as seguintes características:

• Curva de Phillips: !$ u = u_n - a( \pi - \pi^e)\,\,\,\,\,\,\,0< a< \infty !$

• Função de perda social: !$ L = u^2 + b ( \pi - \pi^M)^2\,\,\,\,\,\,\,0 !$

Em que !$ u,u_n, \pi, \pi^e !$ e !$ \pi^M !$ são, respectivamente, as taxas de desemprego, desemprego natural, inflação, inflação esperada e a meta de inflação. A taxa de desemprego natural é positiva. A Autoridade Monetária (A.M.) escolhe diretamente a taxa de inflação, de modo a minimizar a função de perda social, sujeita à restrição imposta pela curva de Phillips. Os agentes privados têm expectativas racionais e a atuação da A.M. pode se enquadrar em dois diferentes casos:

• Caso (I): antes dos agentes privados formarem suas expectativas, a A.M. se compromete crivelmente a gerar certa taxa de inflação p .
• Caso (II): a A.M. escolhe p após os agentes privados terem formado suas expectativas.

Com base nessas informações, julgue a seguinte afirmativa:

Item 1 - No Caso (II), o equilíbrio se caracteriza por !$ u = u_n !$ e !$ \pi = \pi^M + \left ( { \large a \over b} \right) u_n !$;

Julgue a seguinte afirmativa:

Item 1- Em um dado país, a taxa de juros nominal interna é maior que a externa, enquanto que a taxa de juros real interna é menor que a externa. Se valerem a Paridade Descoberta dos Juros e a Equação de Fisher, então a taxa esperada de inflação interna será menor que a externa;

É correta a afirmativa:

Item 2 - !$ \hat{ \theta} !$ de um parâmetro !$ \theta !$ é consistente se, e somente se, !$ \hat{ \theta} !$ é não viesado e a variância de !$ \hat{ \theta} !$ converge para 0 à medida que o tamanho da amostra tende a infinito ;

Julgue a afirmativa:

Item 1- Se !$ f (x,t) e^{ -c^2t}sen (cx) !$ , então !$ { \Large { \partial^2 f \over \partial x^2}} ( x,t) = { \Large { \partial f \over \partial_t}} (x,t) !$ para todo real c;

Text 2

Excerpts from:

Peek-a-boo

Astronomers get some new toys to play
with

FEW scientists believe that the space shuttle has helped their profession. Mostly, it has been used to convey astronauts to a space station that has produced little worthwhile research and to launch satellites that might have been put into orbit more cheaply by old-fashioned, throwaway rockets. But it has done one thing to assist astronomers. It has allowed what is probably their most famous instrument to be repaired and upgraded. That instrument is the Hubble space telescope, which took the picture of the Carina nebula shown above, and has snapped more than half a million other images over the years. Now, as the shuttle programme draws to its close (the final launch will take place next year), Hubble is to be given its last makeover by the crew of Atlantis. On top of that, if the week has gone well, two other satellites intended to probe the universe’s earliest days will have been launched.

The mission to Hubble, which began on May 11th and is planned to last 11 days, will install a wide-field camera that will let the telescope see galaxies previously beyond its reach. Using this, the eager coteries of astronomers who have access to the instrument will be able to observe young, hot stars that glow mainly in the ultraviolet part of the spectrum. They will also be able to see the first stars and galaxies that formed in the universe, which are now so old and distant that their light has been relegated to the infra-red part of the spectrum by the “red shift” caused by the universe’s expansion. These wonders can be observed only from space, because ozone and water in the Earth’s atmosphere absorb light at those wavelengths.

(...)

(From The Economist print edition, May 14^th 2009)

We can infer from the text that

Item 2 - young, hot stars are chiefly seen in the infra-red part of the spectrum;