Attach workings please.

Please use R studio, Thank you.

2. The probability of a student passing statistics is known to be 0.41; and the probability of a student passing chemistry is known to be 0.55. 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

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 Y2.4. Consider two independent random variables X and Y which have both Gaussian probability density func- tions. Let X - N(ux, 03 ) and Y ~ N(uy, of). 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. W10 points X and Y have a bivariate Gaussian distribution if their joint density function is given by: 1 f (x, y) = + (y - Hy ) 2 270 xy V1 - p2 exp C - Hx 1 2 2p(x -Hx) ( y - My) 2(1 - p2) Ox Oxy Oy where -co 0, and oy > 0. p is the correlation between the random variables X and Y. Prove that X has a Gaussian distribution with mean /, and variance o2. Similarly, show that Y has a Gaussian distribution with mean My and variance on