Continuous Random Variables
Let \(X\) and \(Y\) have the joint probability density function: \[f(x,y)=\frac{1}{49}xy\] for \(0\leq x\leq2\) and \(0\leq y\leq7.\) Find the expected value of \(3X.\)
Let \(X\) and \(Y\) have the joint probability density function: \[f(x,y)=k(x+y)\] for \(0\leq x\leq10\) and \(0\leq y\leq1,\) where \(k\) is a constant. Find the marginal probability \(P(1\leq X\leq2).\)
Let \(X\) and \(Y\) have the joint probability density function: \[f(x,y)=k\] for \(0\leq x\leq5\) and \(0\leq y\leq8,\) where \(k\) is a constant. Find the covariance between \(X\) and \(Y.\)
Let \(X\) and \(Y\) have the joint probability density function: \[f(x,y)=kxy\] for \(0\leq x\leq4\) and \(0\leq y\leq3,\) where \(k\) is a constant. Find \(P(X>Y).\)
Which of the following CANNOT be the joint probability density function of two continuous random variables \(X\) and \(Y?\)
(A): \(f(x,y)=1\) for \(0\leq x,y\leq1.\)
(B): \(f(x,y)=x+y\) for \(0\leq x,y\leq1.\)
(C): \(f(x,y)=xy\) for \(0\leq x,y\leq1.\)
(D): \(f(x,y)=e^{x+y}\) for \(0\leq x,y\leq\ln2.\)