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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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