Let X and Y have joint PDFfX,Y (x, y) = 3x^2y + 3xy^2 if 0 < x < 1, and 0 < y < 1                    0 otherwise.a)  Find the covariance of X and Y .b)  Find Var(X).c)  Find the correlation of X and Y  (X and Y have the same variance.)

Cov(X,Y) 1) The correlation of two random variables X and Y, usually written as p(X,Y)= Var (X)Var (Y) where Cov(...) is the covariance and Var(.) is the variance of random variables. Let X be a uniform random variable on the interval [0; 1], and let Y = X2. Find the correlation between X and Y.

Use partial derivatives to obtain the formula for the best least-squares fit to the data points. (4,7), (2,3), (3,14) y = (Simplify your answer. Use integers or decimals for any numbers in the expression.)

Calculate the residual of P = (1.2, 2.4) with respect to the line 3x + 4y = 12.

Question 2. Show that, for a simple linear regression, R² = ry, where R² is the coefficient of determination and rxy is the sample correlation between Y and X. Hint: Σ" (Υ - Y)2 Σ1(Y₁-Y)²¹ SxY √SXXSYY' where Sxy = 1 (X; — X)(Y; — Y), Sxx = ₁1(X; - X)², Syy = ₁₁(Y₁ - Y)². R² = "XY =

Q4) The probability mass function of Y is f(y) = y/6 for y=1,2,3,4. Find the variance of Y.

Calculate correlation and probable error between physical parameters (temperature and humidity) Number of families(x)=2 Number of sites(n)= 1 Temperature(y) in autumn= 24.1 Humidity(y) in autumn= 35 20 1 19 Calculate correlation and probable error between physical parameters (temperature and humidity) Using the following formula coefficient of correlation (r). r = n([xy)-([x) (Ey) V[n(Ex 2)-(Ex) 2][n(Ey2)-([y) 2] I