Question

Performance by a student in this course depends on his/her marks obtained in the quizzes and his/her marks obtained in the exams. Let us denote the following random variables:

X: marks obtained in the quizzes

Y : marks obtained in the exams

Z: performance in the course

Assume that the mean and standard deviation of Z are µZ and σZ, respectively. Further assume that X and Y are related to Z as

X = aZ

Y = b + cZ

where a, b, and c are constants. Mean and standard deviation of X are µX and σX, respectively. Mean and standard deviation of Y are µY and σY , respectively.

(a) Show that E[XY ] = abµz + acσ^2z + acµ^2z.

(b) Show that σXY = acσ^2z

(c) Calculate ρXY and comment on the correlation between the marks obtained in the quizzes and the marks obtained in the exams

Answer 1