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Continuous Random Variables

When will that bus finally arrive? How hot is it going to be outside today? These any many other real-world values can be modeled by continuous random variables.

Joint Probability Distribution

         

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 \(0\leq x,y\leq\ln2.\)

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