Take two orthogonal vectors u, v ε R". u v, and let X = (Xi, , X,) be a vector of n iid standard Gaussians. Xi ~ N(0, 1), yl E「r). Let llr-〈II, X〉 and vr-(v, X〉, Are ux and vr independent? Hin: First try to see if they are correlated; you may use the fact that jointly normal random variables are independent iff. they are uncorrelated.

In a coastal town, the humidity, measured in grams of water per kilogram of air, increases by 43% for every 1 degree celsius increase in temperature. a scientist observes that the humidity on monday morning is 5.75 grams per kilogram. he wants to know how much the temperature must increase for the humidity to reach at least 49.17 grams per kilogram. let t represent the increase in temperature, in degrees celsius, from monday morning's observation. write an inequality to represent the situation, and use it to determine the desired temperature increase.