Determine the conditional probability distribution of Y given that X = 1. Where the joint probability density function is given by f(x,y)=1/64xy for 0 < x < 4 and 0 < y < 4.
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- b) Let Z₁ =*-~N(0,1), and W₁=~N(0,1), for i=1,2,3,...,10, then: i) State, with parameter(s), the probability distribution of the statistic, T = - 10 ii) Find the mean and variance of the statistic T = iii) Calculate the probability that a statistic T = Z₁ Σας Ζα + W₁ is at most 4.Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.Suppose that Y is a random variable that takes on only integer values 1, 2, ... and has distribution function F(y). Show that the probability function p(y) = P(Y = y) is given by the following. F(1), p(y) = y = 1, Fly) - F(y - 1), y = 2, 3, ... Since Y takes on only non-negative integer values, when y > 1 we know P(Y = y) = p(y) can be written as p(y) = --Select- v. By definition F(y) = P(Y?y). And so p(y) = Select-- v. Finally, since Y 2 1 we know p(1) = --Select-- v. And so, we know p(1) = Select--
- What does the memoryless property refer to? An event cannot occur if it has already occured within a recent specified time period. The distribution of the time until an event does not depend on how much time has already passed. The occurrence of an event is independent of the number of times that event has already occurred. None of the above. Suppose X~ GAM (a, 3). Which of the following statements are true? Select all that apply. 3 is a location parameter. E[X]=a3 a is a location parameter. 3 is a shape parameter. 3 is a scale parameter. a is a shape parameter. a is a scale parameter. For a > 1, the probability density function (pdf) is monotonic decreasing. For a < 1, the probability density function (pdf) is monotonic decreasing. None of the above.Suppose that X and Y have the following joint probability distribution y 2 4 1 ГО.1 0.15 = x3 |0.2 5 Lo.1 P(X, Y) 0.3 0.15] a) Evaluate the marginal distribution of X and Y. b) Find P(Y/X) and P(X/Y). c) Find P(Y=2/X=3). d) Find µx , Hy, og, oý and oxY.Suppose that Y is a random variable that takes on only integer values 1, 2, ... and has distribution function F(y). Show that the probability function p(y) = P(Y = y) is given by the following. y = 1, p(v) = Fv) - Fly - 1), y = 2, 3, ... Since Y takes on only non-negative integer values, when y > 1 we know P(Y = y) = p(y) can be written as p(y) = -Select-- ♥. By definition F(v) = P(Y ? y). And so p(y) = -Select--- v. Finally, since Y 21 we know p(1) = -Select--v. And so, we know p(1) = ---Select-
- 2. If U ~ Uniform(1,3), find the probability density function of . .Let X ~ N(0, 2) and Y ~ covariances Cov(X,Y) and Cov(Y, X +Y). Exp(A = 3) be two uncorrelated random variables. Find theThere are only two states of the world, when a person is well with probability (1-p) and ill with probability p, where (1-p)= 1/3 and p = 2/3. Consider Adam who has utility function U = (Y1, Y2,1 – p,p) = Y,"-P)Y?, where Y; is the income and i =1 is well and i = 2 is ill. When Adam is well, he earns $1000, but when he is ill, he losses $300 in health expenditures and earnings such that Y2 = $700. What is the maximum total premium that Adam is still willing-to-pay? (1-р).
- Find k if the joint probability distribution of X, Y, andZ is given by f(x, y, z) = kxyzfor x = 1, 2; y = 1, 2, 3; z = 1, 2.The common probability function of X and Y random variableslike thisa. Find the value of Cov (X, Y).b. U random variable defined as U = X + YFind the probability function.Suppose that three random variables x,„X,„X, fom a random sample from the uniform distribution on the interval [0,2], and they are independent. Determine the value of E(2X, –3X, +X,-4). Var(2X,–3X,+X, -4) and E(2x, –3X, +X, -4)°]