Chapter 8

Two cards are selected at random, without replacement, from a standard deck. Let \(X\) be the number of aces and \(Y\) be the number of spades. Under the usual assumptions, determine the joint distribution and the marginals.

Two positions for campus jobs are open. Two sophomores, three juniors, and three seniors apply. It is decided to select two at random (each possible pair equally likely). Let \(X\) be the number of sophomores and \(Y\) be the number of juniors who are selected. Determine the joint distribution for the pair \(\\{X, Y\\}\) and from this determine the marginals for each.

A die is rolled. Let \(X\) be the number that turns up. A coin is flipped \(X\) times. Let \(Y\) be the number of heads that turn up. Determine the joint distribution for the pair \(\\{X, Y\\}\). Assume \(P(X=k)=1 / 6\) for \(1 \leq k \leq 6\) and for each \(k, P(Y=j \mid X=k)\) has the binomial \((k, 1 / 2)\) distribution. Arrange the joint matrix as on the plane, with values of \(Y\) increasing upward. Determine the marginal distribution for \(Y\). (For a MATLAB based way to determine the joint distribution see Example 7 (Example 14.7: A random number \(N\) of Bernoulli trials) from "Conditional Expectation, Regression")

The pair \(\\{X, Y\\}\) has the joint distribution (in m-file npr08_08.m (Section 17.8.39: npr08_08)): $$P(X=t, Y=u)$$ $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline \mathrm{t}= & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 \\ \hline \mathrm{u}=12 & 0.0156 & 0.0191 & 0.0081 & 0.0035 & 0.0091 & 0.0070 & 0.0098 & 0.0056 & 0.0091 & 0.0049 \\ \hline 10 & 0.0064 & 0.0204 & 0.0108 & 0.0040 & 0.0054 & 0.0080 & 0.0112 & 0.0064 & 0.0104 & 0.0056 \\ \hline 9 & 0.0196 & 0.0256 & 0.0126 & 0.0060 & 0.0156 & 0.0120 & 0.0168 & 0.0096 & 0.0056 & 0.0084 \\ \hline 5 & 0.0112 & 0.0182 & 0.0108 & 0.0070 & 0.0182 & 0.0140 & 0.0196 & 0.0012 & 0.0182 & 0.0038 \\ \hline 3 & 0.0060 & 0.0260 & 0.0162 & 0.0050 & 0.0160 & 0.0200 & 0.0280 & 0.0060 & 0.0160 & 0.0040 \\ \hline-1 & 0.0096 & 0.0056 & 0.0072 & 0.0060 & 0.0256 & 0.0120 & 0.0268 & 0.0096 & 0.0256 & 0.0084 \\ \hline-3 & 0.0044 & 0.0134 & 0.0180 & 0.0140 & 0.0234 & 0.0180 & 0.0252 & 0.0244 & 0.0234 & 0.0126 \\ \hline-5 & 0.0072 & 0.0017 & 0.0063 & 0.0045 & 0.0167 & 0.0090 & 0.0026 & 0.0172 & 0.0217 & 0.0223 \\ \hline\end{array}$$ Determine the marginal distributions. Determine \(F_{X Y}(10,6)\) and \(P(X>Y)\)

Data were kept on the effect of training time on the time to perform a job on a production line. \(X\) is the amount of training, in hours, and \(Y\) is the time to perform the task, in minutes. The data are as follows (in m-file npr08_09.m (Section 17.8.40: npr08_09)): $$P(X=t, Y=u)$$ $$\begin{array}{|l|l|l|l|l|l|}\hline \mathrm{t}= & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline \mathrm{u}=5 & 0.039 & 0.011 & 0.005 & 0.001 & 0.001 \\\\\hline 4 & 0.065 & 0.070 & 0.050 & 0.015 & 0.010 \\ \hline 3 & 0.031 & 0.061 & 0.137 & 0.051 & 0.033 \\\\\hline 2 & 0.012 & 0.049 & 0.163 & 0.058 & 0.039 \\ \hline 1 & 0.003 & 0.009 & 0.045 & 0.025 & 0.017 \\\\\hline\end{array}$$ Determine the marginal distributions. Determine \(F_{X Y}(2,3)\) and \(P(Y / X \geq 1.25)\).

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. \(f_{X Y}(t, u)=1\) for \(0 \leq t \leq 1,0 \leq u \leq 2(1-t)\) \(P(X>1 / 2, Y>1), \quad P(0 \leq X \leq 1 / 2, Y>1 / 2), P(Y \leq X)\)

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. \(\begin{aligned} f_{X Y}(t, u)=4 t &(1-u) \text { for } 0 \leq t \leq 1,0 \leq u \leq 1 \\ & P(1 / 21 / 2), \quad P(X \leq 1 / 2, Y>1 / 2), \quad P(Y \leq X) \end{aligned}\)

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. \(\begin{aligned} f_{X Y}(t, u)=& 4 t(1-u) \text { for } 0 \leq t \leq 1,0 \leq u \leq 1 \\ & P(1 / 21 / 2), \quad P(X \leq 1 / 2, Y>1 / 2), \quad P(Y \leq X) \end{aligned}\)

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. \(f_{X Y}(t, u)=\frac{1}{8}(t+u)\) for \(0 \leq t \leq 2,0 \leq u \leq 2\) \(P(X>1 / 2, Y>1 / 2), \quad P(0 \leq X \leq 1, Y>1), P(Y \leq X)\)

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. \(f_{X Y}(t, u)=4 u e^{-2 t}\) for \(0 \leq t, 0 \leq u \leq 1\) \(P(X \leq 1, Y>1), \quad P(X>0.5,1 / 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. \(f_{X Y}(t, u)=\frac{3}{88}\left(2 t+3 u^{2}\right)\) 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 \(0 \leq u \leq 1\) \(P(1 / 2 \leq X \leq 3 / 2, Y \leq 1 / 2)\)