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- Q1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=0endent. 3.60 The joint probability density function of the ran- dom variables X, Y, and Z is ry=", 0 }, 2 < Z < 3); (d) P(0 < Z < 2 | X = , Y = }). 75°F Mostly cloudyLet z = f(u, v) where u = x*y, v = =, and f is twice differentiable with fu(4, 2) = 4, f.(4, 2) = -5, fuu(4, 2) = -1, fur (4, 2) = fou(4, 2) = 3 and fov (4, 2) = 2. Find %3| %3D -|(z.y)=(2,1)- drðy
- = 1. (Review) (a) Suppose that X ~ Cauchya and c> 0. Show that Y cX Cauchyca (Hint: This is change of variable problem. First find the cdf for Y, then find the pdf by differentiating) (b) If X₁, ..., Xn are i.i.d. and X; ~ Cauchya, then Sn~ Cauchya (The point: Unlike in the various versions of the law of large numbers, Sn does not converge to a constant in any sense).4. What are the median, and mode, respectively, for the density function f(x) = Ta (1 + x²)'2. Consider a decision maker with utility function: u (w) = a - Be r", a, 3,r > 0 where w denotes the total wealth of the decision maker. Suppose she can accept or reject a lottery. If she rejects the lottery she keeps a certain wealth level wo. If she accepts the lottery she gains or loses the random amount e which is distributed according to a density function f (e) and her total (random) wealth is wo +e. Does the decision to accept or reject the lottery depend on her initial wealth wo? (a) No (b) Yes (c) Uncertain (d) Cannot be inferred from the given information.
- 2) d Let the value of your When Anto of damage to your car where X is a random density function f(x) a) Find 'a' function. b) What where You have This means full amount of damages, then you pay insurance X let Y damage. damage Find the give the of car be M = £2000 an accident the is X pounds, variable with 2 (M-x)/a, 10 positive with an if X ≤ M/2 then X. you get M a insurance + company M/2 pounds covers M/2 pounds. be the So that f(x) is the probability that to your car will be Cumulative distribution of X. formula Y as 0 ≤ x ≤ M₁ x M constant. M/2 pounds excess. you pay the while if X > M/2 and the the remaining amount you pay for the amount probability a probability density the cost or less than £1000. function that expresses a3. Let X be a scalar random sample from the following density f(x) = 2(0-x) 82 Construct a (1-a) Cl for 0. for 0 ≤ x ≤ 0asysb, If the density function of Y is f(y)={ b-a' 0, Otherwise. then the variance of Y is equal to Select one: (b-a) a. 12 (a+b? Ob. 12 a+b
- 24. Consider a distribution with density on the interval [0, 2]. Let the probability density function (pdf) for this distribution be the following: fy(u) = % on [0, 2] 2 (i) Draw/plot the pdf fy(y) vs. y for the interval [0, 2]. (ii) Determine the Cumulative Distribution Function (CDF), F, (y). (iii) Draw/plot the CDF F,(y) vs y for the interval [0, 2]. (iv) Determine the expected value of the Random Variable (RV) y, i.e., E [y].b) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0SEE MORE QUESTIONS