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.

Let X1, X2, . . . are independent indicator variables with different probabilities of success. That is, P(Xi = 1) = pi. Define Yn = X1 + X2 + . . . + Xn. Find E(Yn), V ar(Yn) and coefficient of variation of Yn

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

Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.

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.)

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