91Ó°ÊÓ

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=\frac{3}{88}\left(2 t+3 u^{2}\right)\) for \(0 \leq t \leq 2,0 \leq u \leq 1+t\) \(F_{X Y}(1,1), P(X \leq 1, Y>1), P(|X-Y|<1)\)

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=12 t^{2} u\) on the parallelogram with vertices \((-1,0), \quad(0,0), \quad(1,1), \quad(0,1)\) $$P(X \leq 1 / 2, Y>0), \quad P(X<1 / 2, Y \leq 1 / 2), P(Y \geq 1 / 2)$$

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(\begin{aligned} f_{X Y}(t, u)=\frac{24}{11} t u \text { for } 0 \leq t & \leq 2,0 \leq u \leq \min \\{1,2-t\\} \\ & P(X \leq 1, Y \leq 1), \quad P(X>1), P(X

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=\frac{3}{23}(t+2 u)\) for \(0 \leq t \leq 2,0 \leq u \leq \max \\{2-t, t\\}\) \(P(X \geq 1, Y \geq 1), \quad P(Y \leq 1), \quad P(Y \leq X)\)

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=\frac{12}{179}\left(3 t^{2}+u\right),\) for \(0 \leq t \leq 2,0 \leq u \leq \min \\{2,3-t\\}\) \(P(X \geq 1, Y \geq 1), P(X \leq 1, Y \leq 1), P(Y

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=\frac{2}{13}(t+2 u)\) for \(0 \leq t \leq 2,0 \leq u \leq \min \\{2 t, 3-t\\}\) \(P(X<1), \quad P(X \geq 1, Y \leq 1), \quad P(Y \leq X / 2)\)

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=I_{[0,1]}(t) \frac{3}{8}\left(t^{2}+2 u\right)+I_{(1,2]}(t) \frac{9}{14} t^{2} u^{2}\) for \(0 \leq u \leq 1\) \(P(1 / 2 \leq X \leq 3 / 2, Y \leq 1 / 2)\)