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Use the following information for all steps in this problem. Input all answers to three decimals places and do not input your answers as a percentage.

Alec is taking both a Statistics course and a Geography course this semester. The probability that he passes his Statistics course is 0.65. The probability that he passes his Geography course is 0.92. These two events (passing Statistics and passing Geography) are independent.

1. What is the probability he passes both classes?

2. What is the probability he passes , but fails the other?

3. What is the probability he passes at least?

4. What is the probability he passes neither class?

5. What is the probability he passes Statistics and fails Geography?

6. What is the probability he passes Geography and fails Statistics?

Let X = [X1 X2 denote a bivariate normal (Gaussian ) ran- dom vector. Assume EX = 0 and EXXT Define Y1 = X1 + X2 and Y2 = -X1 + X2 1. Find the joint distribution of Y1 and Y2; find the marginal distributions of Y1 and Y2. 2. Find the conditional density of X1, given Y1; find the conditional den- sity of X1, given Y2. 3. Find the conditional mean and variance of X1, given Y1; find the conditional mean and variance of X1, given Y2.Bivariate Distributions What is the bivariate probability density function (pdf) and its properties? Marginal, joint, and conditional distributions Expected values and variance calculation in bivariate distributions Marginal, joint, and conditional expectations CovarianceY1 = Xisgn (X2), Y2 = X2 sgn (X1), where 1 a > 0, sgn (x) = -1 x50. (a) Find the CDF Fy,(y1) in terms of the $(.) function. (b ) Show that Y and Yo are both Gaussian random variables. Are Y1 and Y2 bivariate Gaussian ran- dom variables?(a) a: has mean 2 and standard deviation 1, y has mean 4 and standard deviation 0.5, and the corre- lation between m and y is 70.5. (b) a: has mean 1 and standard deviation 1, and y is equal to 3