Question Let $C$ be any collection of subsets of a set $X$. Let $A(C)$ the algebra generated by $C$. Prove that the $\sigma$-algebra generated by $C$ and $A(C)$ are the same. Here is my idea of a proof, Proof: I would like to show that one is a subset of another and vice-versa. Let $\sigma(C)$ be the $\sigma$-algebra generated by $C$ and $\sigma [A(C)]$ be the $\sigma$-algebra generated by $A(C)$. Since $C\subset A(C)$ and $A(C)\subset\sigma [A(C)]$ then $C\subset \sigma [A(C)]$. $\sigma(C)$ is the smallest $\sigma$-algebra containing $C$ thus $\sigma(C)\subset\sigma [A(C)]$. For the other way, note that $C\subset \sigma(C)$. Since $\sigma(C)$ is also an algebra thus $A(C)\subset \sigma(C)$. $\sigma [A(C)]$ is the smallest algebra containing $[A(C)]$ thus $\sigma [A(C)]\subset \sigma(C)$. Is my proof correct? In the second part I assumed that $A(C)$ is the smallest algebra containing $C$ that is why I got $A(C)\subset \sigma(C)$. I can't find the definition of "the algebra generated by $C$" anywhere. The definition I have is that the $\sigma$-algebra generated by a set $C$ is the smallest $\sigma$-algebra that contains $C$. So I assumed that "the algebra generated by $C$" is the smallest algebra that contains $C$. Am I right in assuming this? I hope somebody could clarify this for me. Also, what is the smallest algebra containing a set $C$? Is it $\{\emptyset,C\}$? So is $\{\emptyset,C\}$ the algebra generated by $C$? Thank you!

hydrology a) Draw the basin boundary on the given topographic map taking basin outlet as point E. Brook Bros Upper RO 1900 1584 Commons 1650- 7600 Brook 0066 b) For the design of a dam at the outlet with an economic life of 200 years, the accepted risk can be taken as 6%. Determine the return period of its design discharge. Risk=(1-9")

(1 point) Determine the average rate of change of the following function between the given values of the variable: f(x) = x4 - - X; x = -2, x = 3 Average rate of change = .

Question Suppose three point masses are placed in the xy-plane as follows (assume coordinates are given in meters): mi = 5 kg, placed at (-2, -3) m2 = 3 kg, placed at (2,3) 'm3 = 2 kg, placed at (-3, -2) Find the center of mass of the system. Write your answer as an ordered pair Provide your answer below:

let f(x)= x^2-5, g(x)= 2x^3+x and p(x)= f(x)g(x). a) computedp/dx b) compute f'(x)g'(x)

10. A box is lying on top of a trailer, which is parked on the road. According to Newton's third law, what is the reaction force to the gravitational pull of the box on the trailer? The normal force of the trailer on the box The gravitational pull of the trailer on the box The gravitational pull of the trailer on Earth The normal force of the road on the trailer The weight of the trailer The normal force of the trailer on the road The gravitational pull of the box on Earth The weight of the box

Suppose that a company is developing a new touch-screen cell phone. Historically, 70% of their new phones have resulted in high consumer demand, whereas 30% have resulted in low consumer demand. The company has the decision of choosing between two models with different features that require different amount of investment and have different sales potential. Estimated profit/loss are shown in the payoff table below. State of Nature High demand, ($) Low demand, ($) Model 1 300,000 -40,000 Model 2 275,000 -15000 Now suppose the firm conducts a market research study to obtain sample information and better understand the nature of consumer demand. The market research cost $10,000 Analysis of past market research studies, conducted prior to introducing similar products, has found that the survey response will be either high or low. There is a 69% chance that the survey will return a high survey response (results) and there is a 31% chance that the survey will result in lo-demand response (results). The relevant conditional probabilities are as follows: P (high consumer demand, high survey response = 0.913) P (low consumer demand, low survey response = 0.087) P (high consumer demand, high survey response = 0.226) P (low consumer demand, low survey response = 0.774) Calculate the expected value of sample information (EVSI). Develop a decision tree and analyze using QM for Windows. Determine the optimal solution.