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Find the mean perimeter.

1.  The four sides of a picture frame consist of two pieces selected from a population whose mean length is 30 cm with standard deviation 0.1 cm, and two pieces selected from a population whose mean length is 45 cm with standard deviation 0.3 cm

a.  Find the mean perimeter.

b.  Assuming the four pieces are chosen independently, find the standard deviation of the perimeter.

 

2.   A gas station earns $2.60 in revenue for each gallon of regular gas it sells, $2.75 for each gallon of midgrade gas, and $2.90 for each gallon of premium gas. Let X1 , X2 , and X3 denote the numbers of gallons of regular, midgrade, and premium gasoline sold in a day. Assume that X1 , X2 , and X3 have….

What is the standard deviation of the total length?

1.  A certain commercial jet plane uses a mean of 0.15 gallons of fuel per passenger-mile, with a standard deviation of 0.01 gallons.

a.  Find the mean number of gallons the plane uses to fly 8000 miles if it carries 210 passengers.

b.  Assume the amounts of fuel used are independent for each passenger-mile traveled.

c.   Find the standard deviation of the number of gallons of fuel the plane uses to fly 8000 miles while carrying 210 passengers.

d.  The plane used X gallons of fuel to carry 210 passengers 8000 miles.

e.  The fuel efficiency is estimated as X/(210 × 8000). Find the mean of this estimate.

f.     Assuming the amounts of fuel used are independent for each passenger-mile, find the standard deviation of the estimate in….

Find the marginal probability mass function pY (y).

1.  The number of bytes downloaded per second on an information channel has mean 10 5 and standard deviation 10 4 . Among the factors influencing the rate is congestion, which produces alternating periods of faster and slower transmission. Let X represent the number of bytes downloaded in a randomly chosen five-second period.

 

a.  Is it reasonable to assume that μx = 5 × 10 5?

b.  Explain. Is it reasonable to assume that ? Explain.

 

2.  Refer to Exercise 1.

a.  Find the marginal probability mass function pX (x).

b.  Find the marginal probability mass function pY (y).

c.   Find µX

d.  Find µY .

e.  Find σX .

f.     Find σY .

g.  Find Cov(X, Y).

h.  Find ρX,Y .

i.      Are X and….

Find the conditional probability mass function pY|X (y | 2)

1.  Refer to Exercise 1.

a.  Find the conditional probability mass function pY|X (y|0).

b.  Find the conditional probability mass function pX|Y (x|1).

c.   Find the conditional expectation E(Y | X = 0).

d.  Find the conditional expectation E(X | Y = 1).

 

2.  Refer to Exercise 4.

a.  The total number of assemblies that fail to meet specifications is X + Y.

b.  Find µX+Y .

c.   Find σX+Y .

d.  Find P(X + Y = 3).

 

3.  Refer to Exercise 4.

a.  Find the conditional probability mass function pY|X (y | 1)

b.  Find the conditional probability mass function pY|X (y | 2)

c.   Find the conditional expectation E(Y | X = 1).

d.  Find the conditional expectation E(X | Y = 2).

 

Find the conditional probability mass function pY\X (y | 4)

1.  Refer to Exercise 4. Assume that the cost of repairing an assembly whose clearance is too little is $2, and the cost of repairing an assembly whose clearance is too much is $3.

a.  Express the total cost of repairs in terms of X and Y.

b.  Find the mean of the total cost of repairs.

c.   Find the standard deviation of the total cost of repairs.

 

2.   Refer to Exercise 9.

a.  Find µX+Y

b.  . Find σX+Y .

c.   Find P(X + Y = 5)

 

3.  Refer to Exercise 9.

 

a.   Find the conditional probability mass function pY\X (y | 4)

b.   Find the conditional probability mass function pY\X (x | 3)

c.    Find the conditional expectation E(Y | X = 4).

….

 Find the conditional probability mass function pY|X (y | 3).

1.  Refer to Exercise 12.

a.   Let Z = X + Y represent the total number of repairs needed.

b.  Find µZ .

c.    Find σZ .

d.   Find P(Z = 2).

 

2.   Refer to Exercise 12. Assume that the cost of an engine repair is $50, and the cost of a transmission repair is $100. Let T represent the total cost of repairs during a one-hour time interval.

a.  Find µT .

b.  Find σT .

c.    Find P(T = 250)

 

3.  Refer to Exercise 12.

 

a.   Find the conditional probability mass function pY|X (y | 3).

b.   Find the conditional probability mass function pY|X (x | 1).

c.    Find the conditional expectation E(Y | X = 3).

d.  Find the conditional expectation E(X….

Estimate f, and find the uncertainty in the estimate.

1.  The lens equation says that if an object is placed at a distance p from a lens, and an image is formed at a distance q from the lens, then the focal length f satisfies the equation 1/f = 1/p + 1/q. Assume that p = 2.3 ± 0.2 cm and q = 3.1 ± 0.2 cm.

a.  Estimate f, and find the uncertainty in the estimate.

b.  Which would provide a greater reduction in the uncertainty in f: reducing the uncertainty in p to 0.1 cm or reducing the uncertainty in q to 0.1 cm?

 

 

2.    The pressure P, temperature T, and volume V of one mole of an ideal gas are related by the equation PV = 8.31 T, when P is measured….

Estimate M and find the uncertainty in the estimate.

1.  The Beer-Lambert law relates the absorbance A of a solution to the concentration C of a species in solution by A = MLC, where L is the path length and M is the molar absorption coefficient. Assume that C = 1.25 ± 0.03 mol/cm3 , L = 1.2 ± 0.1 cm, and A = 1.30 ± 0.05.

a.  Estimate M and find the uncertainty in the estimate.

b.  Which would provide a greater reduction in the uncertainty in M: reducing the uncertainty in C to 0.01 mol/cm3 , reducing the uncertainty in L to 0.05 cm, or reducing the uncertainty in A to 0.01?

 

 

2.    In the article “Temperature-Dependent Optical Constants of Water Ice in the Near Infrared: New Results and Critical Review of the….

Estimate y, and find the uncertainty in the estimate.

1.  Refer to Exercise 12 in Section 3.2. Assume that τ0 = 50 ± 1 MPa, w = 1.2 ± 0.1 mm, and k = 0.29 ± 0.05 mm−1 .

a.  Estimate τ, and find the uncertainty in the estimate.

b.  Which would provide the greatest reduction in the uncertainty in τ: reducing the uncertainty in τ0 to 0.1 MPa, reducing the uncertainty in w to 0.01 mm, or reducing the uncertainty in k to 0.025 mm−1?

c.   A new, somewhat more expensive process would allow both τ0 and w to be measured with negligible uncertainty. Is it worthwhile to implement the process? Explain.

 

2.    According to Snell's law, the angle of refraction θ2 of a light ray traveling in a medium of index of refraction n….

Estimate v, and find the uncertainty in the estimate.

1.   The resistance R (in ohms) of a cylindrical conductor is given by R = kl/d 2 , where l is the length, d is the diameter, and k is a constant of proportionality. Assume that l = 14.0 ± 0.1 cm and d = 4.4 ± 0.1 cm.

a.  Estimate R, and find the uncertainty in the estimate.

b.  Your answer will be in terms of the proportionality constant k.

c.   Which would provide the greater reduction in the uncertainty in R: reducing the uncertainty in l to 0.05 cm or reducing the uncertainty in d to 0.05 cm?

 

2.   Refer to Exercise 16. In an experiment to determine the value of k, the temperature T at time t = 10 min is measured to be….