Problem 2 Two machines produce rivets for a factory job. The number of sub-standard rivets per hour by the two machines are random variables, denoted by X1 and X2. The bivariate PMF of X1 and X2, Px,x,(x1,x2), is given in the table below. X2=0 X2=1 X2=2 X2=3 X₁-0 0.07 0.05 0.02 0.01 X₁ =1 0.05 0.16 0.12 0.02 X₁ =2 0.02 0.12 0.17 0.05 X₁ =3 0.01 0.01 0.05 0.07 6:17 ☑ canvas.ucsd.edu 口 ... Page 2 of 4 - ZOOM (a) Derive and plot the marginal PMFs of random variables X₁ and X2, i.e., Px, X₁ and Px, X2 (b) Compute the mean and variance of X2. (c) Determine and plot the conditional PMFs Pxx, xx₁. (d) Compute the conditional mean of X2 given X₁, xx₁ = Ex|x₁ X2 X₁. Then take the expected value of x, x, with respect to X1, Ex, x,x,] to find the unconditional mean of X2, Hx,. Compare with your result in part (b). (e) Compute the conditional variance of X2 given X₁, σx=xx|x|x|x|x, Then take the expected value of Ex,x, XX₁ (i.e., conditional mean square of X2 given X₁) with respect to X₁ to obtain the unconditional mean square of X2, E|✗2. Use the latter and the unconditional mean of X2 obtained in part (d) to compute the unconditional variance of X2, i.e., Ex Exx XX Exxx]² = EX²]-x=0x Compare with your result in part (b). (f) Compute the unconditional (or total variance) of X2 using the following formula provided in the Reader (page 46): Var[R] = Ey [Varxy [R| Y=y]]+ Vary [Exy [R| Y=y]] where R = R(X,Y) Hint: Here, take X=X, and Y=X, and R(X,Y) = R(X,,X2)=x₂ (g) In a 1 hour period, if the first machine produces exactly 2 sub-standard rivets, what is the probability that the second machine produces at least one sub-standard rivet. (h) In a 1 hour period, if the first machine produces at least 2 sub-standard rivets, what is the probability that the second machine produces at least one sub-standard rivet. (i) Are random variables X1 and X2 statistically independent? (j) Compute the covarance and correlation coefficient of X1 and X2. (k) Determine the probability that the numbers of sub-standard rivets produced do not differ by more than 1 between one machine and the other. (1) The factory manager estimated that an older machine, which was replaced, produced Y = X₁+X2 substandard rivets per hour. Derive the marginal PMF of the number of sub-standard rivets per hour Y produced by the older machine. Problem 3 The joint probability density function of two random variables X and Y can be represented as fxy x, y = c x²+xy+y2, 0≤x≤2 and 2 y≤4 0, elsewhere 2 G ...
Problem 2 Two machines produce rivets for a factory job. The number of sub-standard rivets per hour by the two machines are random variables, denoted by X1 and X2. The bivariate PMF of X1 and X2, Px,x,(x1,x2), is given in the table below. X2=0 X2=1 X2=2 X2=3 X₁-0 0.07 0.05 0.02 0.01 X₁ =1 0.05 0.16 0.12 0.02 X₁ =2 0.02 0.12 0.17 0.05 X₁ =3 0.01 0.01 0.05 0.07 6:17 ☑ canvas.ucsd.edu 口 ... Page 2 of 4 - ZOOM (a) Derive and plot the marginal PMFs of random variables X₁ and X2, i.e., Px, X₁ and Px, X2 (b) Compute the mean and variance of X2. (c) Determine and plot the conditional PMFs Pxx, xx₁. (d) Compute the conditional mean of X2 given X₁, xx₁ = Ex|x₁ X2 X₁. Then take the expected value of x, x, with respect to X1, Ex, x,x,] to find the unconditional mean of X2, Hx,. Compare with your result in part (b). (e) Compute the conditional variance of X2 given X₁, σx=xx|x|x|x|x, Then take the expected value of Ex,x, XX₁ (i.e., conditional mean square of X2 given X₁) with respect to X₁ to obtain the unconditional mean square of X2, E|✗2. Use the latter and the unconditional mean of X2 obtained in part (d) to compute the unconditional variance of X2, i.e., Ex Exx XX Exxx]² = EX²]-x=0x Compare with your result in part (b). (f) Compute the unconditional (or total variance) of X2 using the following formula provided in the Reader (page 46): Var[R] = Ey [Varxy [R| Y=y]]+ Vary [Exy [R| Y=y]] where R = R(X,Y) Hint: Here, take X=X, and Y=X, and R(X,Y) = R(X,,X2)=x₂ (g) In a 1 hour period, if the first machine produces exactly 2 sub-standard rivets, what is the probability that the second machine produces at least one sub-standard rivet. (h) In a 1 hour period, if the first machine produces at least 2 sub-standard rivets, what is the probability that the second machine produces at least one sub-standard rivet. (i) Are random variables X1 and X2 statistically independent? (j) Compute the covarance and correlation coefficient of X1 and X2. (k) Determine the probability that the numbers of sub-standard rivets produced do not differ by more than 1 between one machine and the other. (1) The factory manager estimated that an older machine, which was replaced, produced Y = X₁+X2 substandard rivets per hour. Derive the marginal PMF of the number of sub-standard rivets per hour Y produced by the older machine. Problem 3 The joint probability density function of two random variables X and Y can be represented as fxy x, y = c x²+xy+y2, 0≤x≤2 and 2 y≤4 0, elsewhere 2 G ...
Engineering Fundamentals: An Introduction to Engineering (MindTap Course List)
5th Edition
ISBN:9781305084766
Author:Saeed Moaveni
Publisher:Saeed Moaveni
Chapter15: Computational Engineering Tools Matlab
Section: Chapter Questions
Problem 20P
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Question
Transcribed Image Text:Problem 2
Two machines produce rivets for a factory job. The number of sub-standard rivets per hour by the two
machines are random variables, denoted by X1 and X2. The bivariate PMF of X1 and X2, Px,x,(x1,x2), is
given in the table below.
X2=0
X2=1
X2=2
X2=3
X₁-0
0.07
0.05
0.02
0.01
X₁ =1
0.05
0.16
0.12
0.02
X₁ =2
0.02
0.12
0.17
0.05
X₁ =3
0.01
0.01
0.05
0.07
![6:17
☑
canvas.ucsd.edu
口
...
Page
2
of 4
-
ZOOM
(a) Derive and plot the marginal PMFs of random variables X₁ and X2, i.e., Px, X₁ and Px, X2
(b) Compute the mean and variance of X2.
(c) Determine and plot the conditional PMFs Pxx, xx₁.
(d) Compute the conditional mean of X2 given X₁, xx₁ = Ex|x₁ X2 X₁. Then take the expected value
of x, x, with respect to X1, Ex, x,x,] to find the unconditional mean of X2, Hx,. Compare with
your result in part (b).
(e) Compute the conditional variance of X2 given X₁, σx=xx|x|x|x|x, Then take the
expected value of Ex,x, XX₁ (i.e., conditional mean square of X2 given X₁) with respect to X₁ to
obtain the unconditional mean square of X2, E|✗2. Use the latter and the unconditional mean of X2
obtained in part (d) to compute the unconditional variance of X2, i.e.,
Ex Exx XX Exxx]² = EX²]-x=0x
Compare with your result in part (b).
(f) Compute the unconditional (or total variance) of X2 using the following formula provided in the
Reader (page 46):
Var[R] = Ey [Varxy [R| Y=y]]+ Vary [Exy [R| Y=y]] where R = R(X,Y)
Hint: Here, take X=X, and Y=X, and R(X,Y) = R(X,,X2)=x₂
(g) In a 1 hour period, if the first machine produces exactly 2 sub-standard rivets, what is the probability
that the second machine produces at least one sub-standard rivet.
(h) In a 1 hour period, if the first machine produces at least 2 sub-standard rivets, what is the probability
that the second machine produces at least one sub-standard rivet.
(i) Are random variables X1 and X2 statistically independent?
(j) Compute the covarance and correlation coefficient of X1 and X2.
(k) Determine the probability that the numbers of sub-standard rivets produced do not differ by more than
1 between one machine and the other.
(1) The factory manager estimated that an older machine, which was replaced, produced Y = X₁+X2
substandard rivets per hour. Derive the marginal PMF of the number of sub-standard rivets per hour Y
produced by the older machine.
Problem 3
The joint probability density function of two random variables X and Y can be represented as
fxy x, y = c x²+xy+y2, 0≤x≤2 and 2 y≤4
0, elsewhere
2
G
...](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41633170-d1cd-4c8b-a82b-0acddcbfc65d%2Fb1164710-b22e-4dbc-98d1-23a8d7b31a57%2Fjalwd3a_processed.png&w=3840&q=75)
Transcribed Image Text:6:17
☑
canvas.ucsd.edu
口
...
Page
2
of 4
-
ZOOM
(a) Derive and plot the marginal PMFs of random variables X₁ and X2, i.e., Px, X₁ and Px, X2
(b) Compute the mean and variance of X2.
(c) Determine and plot the conditional PMFs Pxx, xx₁.
(d) Compute the conditional mean of X2 given X₁, xx₁ = Ex|x₁ X2 X₁. Then take the expected value
of x, x, with respect to X1, Ex, x,x,] to find the unconditional mean of X2, Hx,. Compare with
your result in part (b).
(e) Compute the conditional variance of X2 given X₁, σx=xx|x|x|x|x, Then take the
expected value of Ex,x, XX₁ (i.e., conditional mean square of X2 given X₁) with respect to X₁ to
obtain the unconditional mean square of X2, E|✗2. Use the latter and the unconditional mean of X2
obtained in part (d) to compute the unconditional variance of X2, i.e.,
Ex Exx XX Exxx]² = EX²]-x=0x
Compare with your result in part (b).
(f) Compute the unconditional (or total variance) of X2 using the following formula provided in the
Reader (page 46):
Var[R] = Ey [Varxy [R| Y=y]]+ Vary [Exy [R| Y=y]] where R = R(X,Y)
Hint: Here, take X=X, and Y=X, and R(X,Y) = R(X,,X2)=x₂
(g) In a 1 hour period, if the first machine produces exactly 2 sub-standard rivets, what is the probability
that the second machine produces at least one sub-standard rivet.
(h) In a 1 hour period, if the first machine produces at least 2 sub-standard rivets, what is the probability
that the second machine produces at least one sub-standard rivet.
(i) Are random variables X1 and X2 statistically independent?
(j) Compute the covarance and correlation coefficient of X1 and X2.
(k) Determine the probability that the numbers of sub-standard rivets produced do not differ by more than
1 between one machine and the other.
(1) The factory manager estimated that an older machine, which was replaced, produced Y = X₁+X2
substandard rivets per hour. Derive the marginal PMF of the number of sub-standard rivets per hour Y
produced by the older machine.
Problem 3
The joint probability density function of two random variables X and Y can be represented as
fxy x, y = c x²+xy+y2, 0≤x≤2 and 2 y≤4
0, elsewhere
2
G
...
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