Show all your workings kindly.

Given that the probability of a student taking a statistics class is 0.83, and the probability of a student taking a calculus class and a statistics class is 0.66, what is the probability of a student taking a calculus class given that the student takes a statistics class?

Round your answer to three decimal places.

4.10 Suppose you are given random variables x and 3; such that 'r " \"(pin 0:) j'h.' "' NU?\" +1611: 0-2} so you have the marginal distribution of x and the conditional distribution of y given it. The joint distribution of (x, y) is bivariate normal. Find the 5 parameters Wm\"? 53' 55'9\") of the bivariate normal. Let x(1) be a Gaussian process in which two random variables are x1 = x(f1) and x2 = x(12) The random variables have variances of of and o and means of m, and my. The correlation coefficient is p = (X - m)(x2 - my)/(0102) Using matrix notation for the / = 2-dimensional PDF, show that the equation for the PDF of x reduces to the bivariate Gaussian PDF as given by Eq. (B-97).4. Consider two independent random variables X and Y which have both Gaussian probability density func- tions. Let X - N(ux, 03) and Y ~ N(ay, o;). Two new random variables are formed through the following linear transformation. Find the joint probability density function of the new random variables Z and W, faw (z, w) =? Hint: Z and W will have bivariate Gaussian distribution as linear transformations do not change the type of distributions. Hence, it will suffice to find the mean and covariance matrix of the new random variables. W1 25 points For some constant c > 0, X and Y are bivariate Gaussian random variables with joint PDF exp ( - (2 1)2/4 (x 1) (y 3)/3+ (y 3)2 /4 fx, y (x, y) = 10/9 87 VC Let W = 2X + Y and R = X +Y. Choose all the true statements. Keep in mind that the number of true statements is random; it's possible that all 10 statements are true. . Note that (z ) is the CDF of the Standard Normal Random Variable Z, and Q (z) is the Standard Normal Complementary CDF. The correlation coefficient PW.R = 0.782. Var W] = 92/3. The constant c = 0.556. P(X + Y > 6) = Q(2/V3). R is Gaussian (4, 48/3). W and R are bivariate Gaussian random variables. The correlation coefficient px Y = 1/3. P(2X + Y