Numerade

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Reputational elasticity is similar to inertia and an equally important concept that recognizes not the momentum factor but rather that reputation is sometimes elastic and wants to return to its original position when something may move it quickly in one direction or another

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(15 pts) There is a 6 x 6 grid consisting of 36 squares. For each square, there is a 1/3 chance that it is infested with termites. Assume that whether a square is infested with termites is independent of whether other squares are infested with termites (i.e., whether the squares are infested with termites is determined by 36 independent flips of a biased coin). Now, you want to build a $b \times b$ house which occupies $b^2$ squares (arranged as a $b \times b$ larger square). You cannot build the house on a square infested with termites. For example, if the squares infested with termites are marked with "X" as follows: XXX XXX XXXXXX XX XX XX XX XXXXXX Then there are 10 possible positions for building a 1 x 1 house, 3 possible positions for building a 2 x 2 house (note that these positions may partially overlap), and 0 possible positions for building a 6 x 6 house. (a) (5 pts) Let X be the number of possible positions for building a 1 x 1 house. Find the probability mass function $p_X(x)$, and the expectation $E(X)$. (b) (5 pts) Let Y be the number of possible positions for building a 6 x 6 house. Find the probability mass function $p_Y(y)$, and the expectation $E(Y)$. (c) (5 pts) Find the probability that there are at least 24 possible positions for building a 2 x 2 house. (Hint: Consider the squares infested with termites. Can there be more than 1 such squares? Where must these squares be?)

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1. A robot starts at (0,0). It can move either (+1,0) or (0,+1), i.e. up or right. For example, it can move to the point (1,2) by applying (+1,0) then move (0,+1) twice. (a) How many ways can the robot get to the point (10,10). (Hint: Start figuring out how many ways the robot can move to (1,1), (2,2),... first). (b) Someone put a boulder at (5,8) that the robot cannot pass through. Now how many paths are there to (10,10)? (c) Now in addition to the boulder at (5,8), there is another boulder at (4,4). How many paths to (10,10) are there now? Hint: INCLUDE some, EXCLUDE others...

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The joint p.d.f. of $ X $ and $ Y $ is given by \[ f(x, y)=\left\{\begin{array}{ll} \frac{6-x-y}{8}, & 0<x<2,2<y<4 \\ 0, & \text { elsewhere. } \end{array}\right. \] Calculate (i) $ \mathrm{P}(\mathrm{X}<1, \mathrm{Y}<3 $ ), (ii) $ \mathrm{P}(\mathrm{X}+\mathrm{Y}<3 $ ), (iii) $ \mathrm{P}(\mathrm{X}<1 \mid \mathrm{Y}=3) $, (iv) $ \mathrm{P}(\mathrm{X}<1 \mid \mathrm{Y}<3) $. (Ans. (i) $ \frac{3}{8} $, (ii) $ \frac{5}{24} $, (iii) $ \frac{5}{8} $, (iv) $ \frac{5}{8} $ )

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what is the amount and frequency of dividend payments on a bond that has a face value of $10,000 and a coupon rate of 8% per year paid quarterly

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$ f(x)=\frac{e^{3 x+4}}{\ln \sqrt[5]{x^{2}-2}} $

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Which concept concerning complete political authority and power is central to federalism?

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(17.1) A U-tube has two immiscible (non-mixing) fluids in it, as illustrated in the figure. Both columns are open to the atmosphere. The density of the fluid in the column on the left is (initially) unknown. The fluid in the right column has density $\rho = 2000 \text{ kg/m}^3$. The heights of the fluids in the two columns are labelled in the figure. Take $g = 10 \text{m/s}^2$ and $P_{atm} = 101300 \text{ Pa}$. Determine the density of the fluid in the left column.

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(b) For Liquids: Sound travels much faster in condensed phases than in most gases. From equation (2-53), we can see that the partial derivative $\left(\frac{\partial P}{\partial V}\right)_T = -\frac{1}{kV}$ (use the inverse rule to get this). Then, $c = \sqrt{\frac{C_P}{C_V}\frac{V}{\kappa M_W}}$ Although the heat capacity ratio is taken as unity for solids and liquids, $\left(\frac{\partial P}{\partial V}\right)_T$ is very large for condensed phases compared with gases. Example 2.13: Given that for water $\kappa_{298K} = 4.5 \times 10^{-10} / bar$. Calculate the speed of sound in water at 298K.

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240V 20 WW 20 WW 1o 20 Vo jn 40A

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benjamin g.