You can use R to evaluate sums and integrals if you can not do it analytically.

Problem 1

Say the random variable X has density \(f(x)=c/x^k\), \(x=1,2,3,...;k=2, 3\text{ or }4\). Find the mean and the variance of X.

Problem 2

Let X be a random variable with density \(f(x)=c/x^a;x>1\), and a any real number for which f is a density. Find the mean and the variance of X.

Problem 3

Let X be a random variable with density \(f(x)=6x(1-x);0<x<1\). Find the skewness and the kurtosis of X.

Problem 4

Say \((X,Y)\) is a random vector with joint density proportional to \(g(x,y)=(x+1)y^x;0<y<1;x=1,2,3\). Find the correlation of X and Y.