A manufactured item is classified as good, a “second,” or

PROBLEM 9E A manufactured item is classified as good, a “second,” or defective with probabilities 6/10, 3/10, and 1/10, respectively. Fifteen such items are selected at random from the production line. Let X denote the number of good items, Y the number of seconds, and 15 ? X – Y the number of defective items. (a) Give the joint pmf of X and Y, f (x, y). (b) Sketch the set of integers (x, y) for which f (x, y) > 0. From the shape of this region, can X and Y be independent?Why or why not? (c) Find P(X = 10,Y = 4). (d) Give the marginal pmf of X. (e) Find P(X ? 11).

PROBLEM 4E Select an (even) integer randomly from the set {0, 2, 4, 6, 8}. Then select an integer randomly from the set {0, 1, 2, 3, 4}. Let X equal the integer that is selected from the first set and let Y equal the sum of the two integers. (a) Show the joint pmf of X and Y on the space of X and Y. (b) Compute the marginal pmfs. (c) Are X and Y independent?Why or why not?

PROBLEM 6E The torque required to remove bolts in a steel plate is rated as very high, high, average, and low, and these occur about 30%, 40%, 20%, and 10% of the time, respectively. Suppose n = 25 bolts are rated; what is the probability of rating 7 very high, 8 high, 6 average, and 4 low? Assume independence of the 25 trials.

Roll a pair of four-sided dice, one red and one black, each of which has possible outcomes \(1, 2, 3, 4\) that have equal probabilities. Let \(X\) equal the outcome on the red die, and let \(Y\) equal the outcome on the black die. (a) On graph paper, show the space of \(X\) and Y. (b) Define the joint pmf on the space (similar to Figure 4.1-1). (c) Give the marginal pmf of \(X\) in the margin. (d) Give the marginal pmf of \(Y\) in the margin. (e) Are \(X\) and \(Y\) dependent or independent? Why or why not? Equation Transcription: Text Transcription: X Y 1,2,3,4

In a smoking survey among boys between the ages of 12 and 17, \(78 \%\) prefer to date nonsmokers, \(1 \%\) prefer to date smokers, and \(21 \%\) don’t care. Suppose seven such boys are selected randomly. Let \(X\) equal the number who prefer to date nonsmokers and \(Y\) equal the number who prefer to date smokers. (a) Determine the joint pmf of \(X\) and \(Y\). Be sure to include the support of the pmf. (b) Find the marginal pmf of \(X\). Again include the support. Equation Transcription: 1% Text Transcription: X Y 78% 1% 21%

Roll a pair of four-sided dice, one red and one black. Let \(X\) equal the outcome on the red die and let \(Y\) equal the sum of the two dice. (a) On graph paper, describe the space of \(X\) and Y. (b) Define the joint pmf on the space (similar to Figure 4.1-1). (c) Give the marginal pmf of \(X\) in the margin. (d) Give the marginal pmf of \(Y\) in the margin. (e) Are \(X\) and \(Y\) dependent or independent? Why or why not? Equation Transcription: Text Transcription: X Y

For each of the following functions, determine the constant \(c\) so that \(f(x, y)\) satisfies the conditions of being a joint pmf for two discrete random variables \(X\) and \(Y\): (a) \(f(x, y)=c(x+2 y), \quad x=1,2, \quad y=1,2,3\). (b) \(f(x, y)=c(x+y), \quad x=1,2,3, \quad y=1, \ldots, x\) (c) \(f(x, y)=c, \quad x\) and \(y\) are integers such that \(6 \leq x+y\) \(\leq 8,0 \leq y \leq 5\) . (d) \(f(x, y)=c\left(\frac{1}{4}\right)^{x}\left(\frac{1}{3}\right)^{y}, \quad x=1,2, \ldots, \quad y=1,2, \ldots\)

Roll a pair of four-sided dice, one red and one black, each of which has possible outcomes 1, 2, 3, 4 that have equal probabilities. Let X equal the outcome on the red die, and let Y equal the outcome on the black die. (a) On graph paper, show the space of X and Y. (b) Define the joint pmf on the space (similar to Figure 4.1-1). (c) Give the marginal pmf of X in the margin. (d) Give the marginal pmf of Y in the margin. (e) Are X and Y dependent or independent? Why or why not?

Let the joint pmf of X and Y be defined by f(x, y) = x + y 32 , x = 1, 2, y = 1, 2, 3, 4. (a) Find fX (x), the marginal pmf of X. (b) Find fY (y), the marginal pmf of Y. (c) Find P(X > Y). (d) Find P(Y = 2X). (e) Find P(X + Y = 3). (f) Find P(X 3 Y). (g) Are X and Y independent or dependent? Why or why not? (h) Find the means and the variances of X and Y.

Select an (even) integer randomly from the set {0, 2, 4, 6, 8}. Then select an integer randomly from the set {0, 1, 2, 3, 4}. Let X equal the integer that is selected from the first set and let Y equal the sum of the two integers.(a) Show the joint pmf of X and Y on the space of X and Y. (b) Compute the marginal pmfs. (c) Are X and Y independent? Why or why not?

Roll a pair of four-sided dice, one red and one black. Let X equal the outcome on the red die and let Y equal the sum of the two dice. (a) On graph paper, describe the space of X and Y. (b) Define the joint pmf on the space (similar to Figure 4.1-1). (c) Give the marginal pmf of X in the margin. (d) Give the marginal pmf of Y in the margin. (e) Are X and Y dependent or independent? Why or why not?

The torque required to remove bolts in a steel plate is rated as very high, high, average, and low, and these occur about 30%, 40%, 20%, and 10% of the time, respectively. Suppose n = 25 bolts are rated; what is the probability of rating 7 very high, 8 high, 6 average, and 4 low? Assume independence of the 25 trials.

A particle starts at (0, 0) and moves in one-unit independent steps with equal probabilities of 1/4 in each of the four directions: north, south, east, and west. Let S equal the eastwest position and T the northsouth position after n steps. (a) Define the joint pmf of S and T with n = 3. On a twodimensional graph, give the probabilities of the joint pmf and the marginal pmfs (similar to Figure 4.1-1). (b) What are the marginal distributions of X and Y?

In a smoking survey among boys between the ages of 12 and 17, 78% prefer to date nonsmokers, 1% prefer to date smokers, and 21% dont care. Suppose seven such boys are selected randomly. Let X equal the number who prefer to date nonsmokers and Y equal the number who prefer to date smokers. (a) Determine the joint pmf of X and Y. Be sure to include the support of the pmf. (b) Find the marginal pmf of X. Again include the support.

A manufactured item is classified as good, a second, or defective with probabilities 6/10, 3/10, and 1/10, respectively. Fifteen such items are selected at random from the production line. Let X denote the number of good items, Y the number of seconds, and 15 X Y the number of defective items. (a) Give the joint pmf of X and Y, f(x, y). (b) Sketch the set of integers (x, y) for which f(x, y) > 0. From the shape of this region, can X and Y be independent? Why or why not? (c) Find P(X = 10, Y = 4). (d) Give the marginal pmf of X. (e) Find P(X 11).