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Human height is well described by a normal random variable. Suppose a simple random sample of two individuals is taken, and $X_1$ and $X_2$ are their heights. The random variables $X_1$ and $X_2$ are i.i.d. normal random variables with common probability density function given by $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$$ Let $F(x)$ be their common CDF. (a) (2 pts) Write an expression for the probability that $X_1 \leq x_1$ and $X_2 \leq x_2$, in terms of the function $F(x)$. (b) (2 pts) The expression in the previous part is called the joint CDF of $X_1$ and $X_2$. If this function is denoted by $G(x_1,x_2)$ then it can be proved that the joint probability density function of $X_1$ and $X_2$, denoted by $g(x_1,x_2)$, is given by $$g(x_1, x_2) = \frac{\partial^2 G}{\partial x_1 \partial x_2}$$ Calculate the joint probability density function. (c) (2 pts) Suppose the heights of the two individuals are 160 cm and 170 cm. Find the value of the joint PDF for this sample, as a function of the population mean $\mu$. (d) (2 pts) The function in part (c) is referred to as a likelihood function. It gives the value of the joint PDF when evaluated at the sampled data points, for different values of the population mean $\mu$. The maximum likelihood estimate for the population mean height is the value of $\mu$ that maximizes this likelihood function. In other words, it is the value of that makes the joint PDF at the sampled data as large as possible. Find this value of $\mu$.

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which of the following is an example of syntrophy -a methanogen produces methane as its final byproduct, an a methylotroph is dependent upon this methane, metabolizing it to wateer and carbon dioxide - infection of a tooth by Stretococcus salivarious and using host carbohydrates - free living Azotobacter vinlandii in the soil fix nitrogen and plants take it up as a source of nitrogen - in a cow's stomach, the cellulose is broken into lipids, but some of theses lipids cannot be taken up and when they leave the cow other organisms can use these extra lipids -streptomyces species releasing an antimicrobial that kills another nearby bacteria

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Evaluate the iterated integral by converting to polar coordinates.\\ $\int_0^1 \int_y^{\sqrt{2 - y^2}} 7(x + y) \, dx \, dy$

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Suppose the demand for a product is given by $D(p) = -8p + 162$. A) Calculate the elasticity of demand at a price of $6. Elasticity = (Round to three decimal places.) B) At what price do you have unit elasticity? (Round your answer to the nearest penny.) Price = $

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The following are the net currency positions of a U.S. FI (stated in U.S. dollars). Currency Assets Liabilities FX Bought FX Sold British pound 24,600 70,000 170,400 321,000 Yen 31,000 20,400 250,000 220,000 Swiss franc 10,200 9,800 8,000 10,800 What is the Fl's net exposure in the Swiss franc? Select one: ? ?. +3,200. ? ?. -2,400. OC. +2,400. OD. -2,800. ? ?. +400.

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A soda-dispensing machine fills cups with amounts of soda that are normally distributed with population standard deviation 0.5 ounces. In a sample of 17 cups, the mean amount of soda per cup was found to be 12.24 ounces. A hypothesis test is conducted at the $alpha = 0.01$ significance level to determine whether the mean amount of cola in all cups filled by the machine is greater than 12 ounces. Find the P-value for this hypothesis test. Round your answer to four decimal places. Answer: P-value =

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A block with a mass of 0.30 kg is attached to a horizontal spring. The block is pulled back from its equilibrium position until the spring exerts a force of 1.0 N on the block. When the block is released, it oscillates with a frequency of 1.5 Hz. How far was the block pulled back before being released? A small earthquake starts a lamppost vibrating back and forth. The amplitude of the vibration of the top of the lamppost is 6.7 cm at the moment the quake stops, and 8.4 s later it is 1.5 cm. A. What is the time constant for the damping of the oscillation? B. What was the amplitude of the oscillation 4.2 s after the quake stopped?

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2. The following figure shows the resource allocation in an operating system. a. Can you please tell if there is a problem in the figure and why? (5 points) b. How can we solve the problem (if any) or potential resource allocation problems (if no problem in the figure) in general? One general solution is sufficient for the answer. (5 points)

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Based on information in the table, the output gap is Cyclical Unemployment 4% Frictional Unemployment 3% Equilibrium Real Federal Fund Rate 2.40% Inflation Rate 2% Structural Unemployment 3.50% RGDP BD 480 million -0.0319 3.19% -0.0428 4.278%

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clinton m.