James and Daniels. They decided to study separately for their exam. The probability that Jack will make an "A" in statistics is 0.6, and the probability that Sam will make an"A" is 0.3. Find the probability that exactly one of Jack or Sam will make an "A"in statistics

5. Let X and Y be two independent Gaussian random variables with means and variances (Xx, of) and (Hy, ( ), respectively. (a) Show that the joint probability density function of X and Y can be written as: + (y - My ) 2 exp Hint: Because X and Y are independent, we have f(x, y) = f(x)f(y). Also, the probability density function of a Gaussian random variable X is: f (x) = exp V270 1 202 (b) Let Z be a bivariate Gaussian distribution such that Z = and covariance matrix E = 0 02 Show that 1 1 I - HE f (2) = f(x, y) = 27 | |1/2 5exp y - Hy. y - HylExample I (Rosenberg [91]). Let f and g be PDFs with corresponding DFs F and G. Also, let (10) h(x, y) = f(x):(y)[1 + a(2F(x) - 1)(2G(y) - 1)]. where ja| ), 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