Denote a bivariate probability density function for two random variable X and Y as f(x,y). Let f(x,y) = x + y for {?0???X???1 } and {?0???Y???1?}; f(x,y) = 0 otherwise.
Using this bivariate probability density function, find the Pr{ X???0.5 ? Y???0.5?}.
Using this bivariate probability density function, find the marginal probability density function for X. Please denote it by fx(x).
1. The basic difference between macroeconomics and microeconomics is: A. microeconomics explores the causes of inflation while macroeconomics focuses on the causes of unemployment. B. microeconomics concentrates on the behaviour of individual consumers and firms while macroeconomics focuses on the performance of the entire economy. C. microeconomics concentrates on individual markets while macroeconomics focuses primarily on international trade. D. microeconomics concentrates on the behaviour of individual consumers while macroeconomics focuses on the behaviour of firms. 2. Scarcity exists because of: A. unlimited wants and limited resources. B. the allocation of goods by prices C. specialization and division of labor. D. the market mechanism. 3. Production in a society is allocation efficient so long as it uses its resources to produce as much as it can. True FalseFind the mean and Auto Correlation function of X (t) = Acos (cut + $) when w and are fixed. and A is a random variable with Normal density function of zero mean and Variance equal to two. Is x(#) a wide sense stationary random process?Q. 6 In this problem we deal with Bivariate Gaussians. Please read the lecture notes available under the 'Bivariate Gaussian RVs' section on Canvas (L16.pdf). The lecture notes summarizes the following facts about bivariate Gaussians: X and Y are bivariate Gaussian if their joint PDF is given by HX - HX fx,y (x, y ) = 2noxGY (1 - Pxy ) 1/2 exp 2(1 -PxY) - 2PxY Ox Ox This PDF is specified by 5 parameters which correspond to the marginal means, variances and correlation, i.e., (X, Y) ~ N(ux, MY, ox, 67, pxY). Some interesting facts about bivariate Gaussians are listed below: . Marginals are Gaussian: X ~ N(ux, Gx ) and X ~ N(ux, 63). . Bivariate Gaussian RVs are uncorrelated iff they are independent: If pyy = 0 then the joint density can be written as fx, y (x,y) = fx(x) fr ())= 2nox -e- ( x- ux )/20% 1 - ()-MY)2 /20% V2noy thus the RVs are independent. . A linear transformation of bivariate Gaussian is also bivariate Gaussian, but with new parameters. . Bivariate Gaussian and conditioning: For bivariate Gaussian RVs (X, Y) the conditional distributions fry and frix are also Gaussian, and are given by, frix (y | x) = fxr (x, y) - exp (y - HYLx (x))2) fx (x) V2noylx 20 yx ~ N( ur x, 0; x ), where E[Y [ X] = Myx (X) = Hy + Pxy ox (X -ux ) and Gyx = of (1 - Pxy). Suppose that X ~ N(0, 1) and Y ~ N(1, 2) are two independent Gaussian random variables. Define random variables U and V as U = X + Y, V = X + ay for some a. 1. What is the joint distribution of U, V in terms of o? 2. Find o (if it exists) such that U and V be independent. 3. Find the conditional distribution fulv (u|5)
