5. Consider the Clayton copula: C(u,v) = (u¯ª +v¯ α>0 -a - a for 0≤u≤1, 0≤v≤1, a. Show that the marginal distributions for u and v are uniform. b. Show that the pdf for this copula is always non-negative. c. Find the lower tail dependence, defined as λ₁ = lim Pr[X ≤ Fx' (u)|Y ≤ Fy'(u)] = u→0+ +0
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- 2. Let dt Find the average %3D value of f on the interval [o, 16. If X is t-distributed, show that X² is F-distributed.3. Let X; N iid U(0, 1) for i e {1, 2, ...,n}. (a) Compute the pdf of Y; = - log(X;), name the distribution that Y; follows and provide any parameter value or values needed. (b) Compute the pdf that W = [I", X, = X1 X2... X, follows. You may use the result given as part of the Example 2.6 video (even though it will only be proved in Chapter 3). (c) Compute E[W] in the two following ways: (i) through exploiting independence of the X, and using EUV] = E[U]E[V] for any independent random variables U and V. (ii) using the pdf of W computed in the previous part.
- 2.Suppose that the return R (in dollars per share) of a stock has the uniform distribution on the interval [-3,7]. Suppose also, that each share of the stock costs $1.50. Let Y be the net return (total return minus cost) on an investment of 10 shares of the stocks. Compute E(Y).1.2. Assume that Billy's driving time to class is uniformly distributed from 15-20 minutes in good weather, and uniformly distributed from 25-30 minutes in bad weather. Let the RV X describe Billy's drive time. If the probability of good weather is 2/3. a) .describe mathematically the PDF of X. b) ..describe graphically the PDF of X. c) .find the probability that Billy's drive time is >18 minutes. d) .find the Expected Value of Billy's drive time, irrespective of the weather.b) c) d)
- Please help asap with this math theory question (: Consider the delta property and the Von Neumann Morgenstern axioms for the following: If you have a risk premium for Gamble A with a probability 1-p of winning r1 and probability p of winning r2. It's y' for Gamble A' with probability p of winning r2 + c, probability 1-p of winning r1+ c. What is y' as a function of y?Please explain how to read the notationsIn commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 10, then it can be shown that the total waiting time Y has the pdf f (y) = = 100 Y 5 10 0 ≤ y ≤ 10 10 ≤ y ≤ 20 a. Sketch a graph of the pdf of Y (do on your own paper, you will not be turning this in. It is for your reference only.) [ f(y)dy=[ -∞ b. Verify that [ f (y) dy = 1 10 120 10 = 1 2 + c. What is the probability that the total waiting time is at most 8 minutes? d. What is the probability that the total waiting time is at most 13 minutes? = 1 e. What is the probability that the total waiting time is between 8 and 13 minutes? f. What is the probability that the total waiting time is either less than 7 minutes or more than 14 minutes?