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Problem 1. Solve the equation via Newton's method, taking as your initial guess . Express your answer to at least 5 decimal places. [If done correctly, this should only take 4 or 5 iterations. If your results do not stabilize that fast, you made a mistake somewhere!]
Problem 2. Consider the differential equation dy/dx = x/y, with the initial condition y(0)=1.
a) Use Euler's method with to approximate y(2).
b) Solve the differential equation to find y(2) exactly.
Problem 3. Suppose the New York Yankees and the San Diego Padres play a 7 game series. Each game is independent, with the Yankees having a 60% chance of winning. What is the probability that the Yankees will win 4 or more of the 7 games? Your written answer should be correct to at least 3 decimal places (e.g. 0.387, or 38.7%)
Problem 4. The rare Tasmanian Guppy (an imaginary fish whose name I just made up) can live for up to a year. Its actual lifespan X (in years) is given by a continuous random variable with pdf
a) Find the corresponding cdf (i.e. F(x)) for all values of x.
b) What is the probability of a randomly selected guppy living between 3 and 6 months?
c) What is the average lifespan of a Tasmanian Guppy.
Problem 5. A computer generates seventy-five independent random numbers, each uniformly distributed between 0 and 2.
a) What is the probability that the first random number is between 0.9 and 1.5?
b) Let S be the sum of the random numbers. What is the mean value of S? What is the standard deviation?
c) What is the probability that S is between 70 and 85?