Show the solutions to the following problems please. 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?

A Bivariate generalized Gaussian distribution given as p(x|M, a, B) = ME ga,s(x/ M-'x), BT(4) go,B(y) = e- +(#) (2# 10)+r Where M is a covariance matrix and unknown. Perfrom Maximum liklihood estimation.Problem 5. (3pts) In class we mentioned that independence of random variables X, Y implies uncorrelatedness but not vice versa. Yet another nice feature of the Gaussian distirbution is that if X and Y are Gaussian and uncorrelated then they are independent. A random vector (X, Y) is said to be standard bivariate Gaussian if its joint density is given by: f(x, y) = (27) (1 + (EXY) 2) x exp (-3 Here E is the covariance matrix given by 1 EXY E = EY X 1 Therefore the inside of the exponent is -1/2(x2 +y+2(EXY)xy). Prove that if (X, Y) are standard bivariate and uncorrlated then they are independent. That is show that the joint denisty f(x, y) factorizes into joint densities for two standard Gaussian ran- dom variables X and Y. You can use that EX = EY = 05. Consider a bivariate distribution with joint pdf given by f(x, y) = cexp ( xy +x2+y2 - 8x - 8y), x,yER, where c is a normalizing constant. (a) By completing the square, or otherwise, show that the conditional distribution of X given Y = y is Gaussian with mean 4/(1 + y') and variance 1/(1 + y?). [3 marks] Similarly, it can be shown that the conditional distribution of Y given X = x is Gaussian with mean 4/(1 + x2) and variance 1/(1 + 2). (b) Using the above results, or otherwise, describe how you could sample from the joint distri- bution f(x, y). [4 marks]Q2.12 Assume a random vector x = X1 follows a bivariate Gaussian distribution: (x | p, >), where H = is the mean vector and _ = po102 is the covariance matrix. Derive the po102 formula to compute mutual information between 1, and 12, i.e. I((1, 12). Q2.13 Given two multivariate Gaussian distributions: M(x | /1, 21 ) and (X | #2, _2), where #, and #2 are the mean vectors, and E, and 22 are the covariance matrices, derive the formula to compute