Attach all working please. If the probability of passing both is known to be 0.35, calculate:

(a) the probability of passing at least one of statistics and chemistry

(b) the probability of a student passing chemistry, given that they passed statistics

(c) Are passing chemistry and statistics independent? Justify

(d) (harder) a group of 33 randomly selected students attend a special seminar on study skills. Of these 33, only 7 fail both. State a sensible null hypothesis, test it, and interpret

Exercise 5.3 (Continuous bivariate random variable) Let p(@, y) = 6(@ -y) be the PDF of a bivariate random variable (X, Y ), where both variables range from zero to one. a. Find CDF. b. Find marginal distributions. c. Find conditional distributions.3.40. Let X and Y be bivariate mean zero Gaussian random variables with variance one and correlation p. Show that there are independent Gaussian random variables U and V, and reals a, b, c and d such that X = aU + bV and Y = cU + dV.Consider the linear model Y = alX1 + a2X2 +b where the vector ( X1, X2) has a bivariate Gaussian distribution with means #1, #2, variances of, 02, and correlation coefficient p12. Consider the vector (X1, Y). One can show (via some involved calculus) that (X1, Y) has a bivariate Gaussian distribution. Determine its two means, two variances, covariance, and correlation coefficient. Remark: In lecture we considered precisely this problem but under the as- sumption that X] and X2 are independent.Explain what it means for two variables to have a bivariate normal distribution. Choose the correct answer below. O A. Two variables have a bivariate normal distribution when for any fixed values of y the corresponding values of x are normally distributed. O B. Two variables have a bivariate normal distribution when the values of x and y are both normally distributed. O C. Two variables have a bivariate normal distribution when for any fixed values of x the corresponding values of y are normally distributed. O D. Two variables have a bivariate normal distribution when for any fixed values of x the corresponding values of y are normally distributed, and for any fixed values of y the corresponding values of x are normally distributed