Recall the covariance of two random variables X and Y is defined as Cov(X, Y) = E[(X − E[X])(Y − E[Y])]. For a multivariate random variable Z (i. e., each index of Z is a random variable), we define the covariance matrix Σ such that Σi j = Cov(Zi , Zj). Concisely, Σ = E[(Z − µ)(Z − µ) > ], where µ is the mean value of the random column vector Z. Prove that the covariance matrix is always positive semidefinite (PSD).

Mr. bernard needs to order more boxes for his 16 inch diameter “super pizza.” use 3.14 to approximate π.

When you arrive at the lake, your friend realises he hasn’t got any swimming trunks. you need to ride to a sports shop. the sports shop is 8350 metres away. how many metres is it to cycle there and back?

Four airplanes carrying a total of 400 passengersarrive at the detroit airport. the airplanes carry, respectively,50, 80, 110, and 160 passengers.a) one of the 400 passengers is selected uniformly at random.let x denote the number of passengers that were on theairplane carrying the randomly selected passenger. find var(x)for x as given in the problemb) one of the 4 airplane drivers is also selected uniformly atrandom. let y denote the number of passengers on the chosen driver’s airplane. find var(y) for y as given in the problem