Exercises

Find the mass of the lamina with the given mass density (Note: the tool for this section will create a lamina of a region).

1. x = 0, x = 1, y = 0, y = 2 2. x = 0,x = y,y = 0,y = 1 m( x,y) = 2 kg per square meter m(x,y) = 2 kg per square meter 3. x = 0,x = p, y = 0,y = sin(x) 4. y = 0,y = 1,x = y,x = y2 m( x,y) = 2x kg per square meter m(x,y) = 2x kg per square meter 5. y = 1-x2, y = x2-1 6. y = 1,y = cosh( x) m( x,y) = x2+y2 kg per square meter m( x,y) = y kg per square meter

Find the center of mass of the lamina of the following regions with the given mass density. (Use 1-6 above)

7. x = 0, x = 1, y = 0, y = 2 8. x = 0,x = y,y = 0,y = 1 m( x,y) = 2 kg per square meter m(x,y) = 2 kg per square meter 9. x = 0,x = p, y = 0,y = sin(x) 10. y = 0,y = 1,x = y,x = y2 m( x,y) = 2x kg per square meter m(x,y) = 2x kg per square meter 11. y = 1-x2,y = x2-1 12. y = 1,y = cosh( x) m( x,y) = x2+y2 kg per square meter m( x,y) = y kg per square meter

Find the centroid of the following regions.

13. x = 0,x = 1,y = 0,y = x 14. x = 0,x = 1,y = 0,y = x2 15. x = -1,x = 1,y = 0,y = 1-| x| 16. y = 0,y = 1,x = 0,x = sin( py) 17. y = 0,y = 1,x = y2,y = x2 18. y = 0,y = 1,x = y2,y = x

Show the following are joint probability density functions over the given sample spaces. Then find the expected values of the random variables X and Y.

19. Sample Space: [0,1] ×[0,5] 20. Sample Space: [0,1] ×[0,4] p( x,y) = 0.2 p( x,y) = 0.25 21. Sample Space: [0,1] ×[0,1]   22. Sample Space: [0,1] ×[0,1]   p( x,y) = x + y     p( x,y) = 4xy 23. Sample Space: 1st Quadrant 24. Sample Space: 1st Quadrant p( x,y) = 0.2e-x-y/5 p( x,y) =  2 p    e-x 1+y2   25. Sample Space: Entire Plane 26. Sample Space: [ 1,¥] ×[ 0,¥] p( x,y) =  1 4 e-| x| -| y| p( x,y) =  1 x e-xy

 

27. A bank operates both a drive-up window and an indoor teller window. On a randomly selected day, let X = the percentage of the work-day that the drive-up window is in use and let Y = percentage of the work-day that the indoor teller window is in use (Note: both X and Y are between 0 and 1 ). Observation over a period of time leads them to the following joint density for X and Y:

p( x,y) = ì ï í ï î     65 x +  65 y2 if  0 £ x,y £ 1 0 otherwise

1. Show that p( x,y) is a joint probability density?
2. What is the probability that two windows combined are in use less than half of the time? (i.e., what is Pr( X + Y  £ 0.5) ?)
3. What is the probability of the drive-up window being used less than the indoor teller window?
4. What is the expected value of X? What is the expected value of Y? Interpret your result.

29. Suppose we are given radioactive substances A and B with decay rates l > 0 and b > 0, respectively. Then the probability that A will decay in the time interval [ s,s+ds] and B will decay in the time interval [ t,t+dt] is given by

lbe-bs-ltdsdt

for s,t ³ 0.

1. What is the probability density?
2. What is the probability that A decays before B (in terms of l and b)?
3. What is the probability that both will have decayed by time T.

30. What are the expected values for the decay times of A and B, respectively, in exercise 29 ( as functions of l and b)? Explain.

31. If X is uniformly randomly distributed in [ 0, 10], then it has a probability density of

p( x) = ì í î 0.1 if  0 £ x £ 1 0 otherwise

What is the probability that two numbers chosen uniformly randomly from [ 0,10] have a sum greater than 10?

32. What is the probability that two numbers chosen uniformly randomly from [ 0,10] have a product greater than 10? (see exercise 31).

33. The probability density for a point (x,y) chosen uniformly randomly in the unit square is

p( x,y) = ì ï í ï î 1     if  0 ≤ x,y ≤ 1 0 otherwise

What is the probability that y will be less than x² in the unit square?

34. A can of mixed nuts contains 1 kg of cashews, peanuts, and almonds. For a randomly selected box, let X and Y represent the weights of the cashews and almonds, respectively, and suppose that the joint density function of these variables is

p( x,y) = ì í î 24xy if 0 £ x £ 1,  0 £ y £ 1,  x+y £ 1 0 otherwise

Show that p(x,y) is a probability density, and then compute the expected values of X and Y. What is the probability that half of the weight of the contents of the can will be cashews?

35. You arrive at a restaurant where the time T between two customers being seated has a density of p(t) = e^-t for t ³ 0. What is the probability that you have less than 10 minutes to wait before being seated if there is one person ahead of you when the person 2 ahead of you is taken to be seated? (Hint: Let S be the time between you and customer ahead of you and let T be the time between the customer ahead of you and the customer just seated.  Then S and T are independent with densities p(s) = e^-s for s ³ 0 and q(t) = e^-t for t ³ 0, respectively).

36. Suppose that on a certain internet server, the time T in seconds between the arrival of two successive packets is exponentially distributed with density

p( t) = 0.01e-0.01t    for t ³ 0

What is the probability of 3 packets being received in less than 50 milliseconds? (Hint: let T be the waiting time between first and second arrival, and let S be the time between the second and third arrival. Assume S and T are independent).

37. Bacteria is growing on a slide which corresponds to [0,20] ×[ 0,5] , where distances are in centimeters. If the density of the bacteria is determined to be

m( x,y) = 1000xy( 20-x) ( 5-y)     bacteria cm2

then about how many bacteria are on the slide?

38. A certain town is overlaid by the unit square with sides of length 1 mile. The population density is then measured to be

m( x,y) = 100xy  people  per  square  mile

What is the approximate population of that town?

39.  Suppose that a region RÈS has mass density m(x,y)

Show that if the center of mass of a laminate of R is (x₁,y₁) and if the center of mass of a laminate of S is ( x₂,y₂) , then the center of mass of a laminate of RÈS has coordinates

x =  x1M1+x2M2 M1+M2 ,        y =  y1M1+y2M2 M1+M2

40. Show that if p₁( x) and p₂( y) are probability densities for x and y, respectively, then

p( x,y) = p1( x) p2( y)

is a probability density on R².

41. Write to Learn: Torque is the tendency of a force to cause an object to rotate about a given axis. In particular, if the force is gravity, then the torque of a laminate of a region R with mass density m( x,y) about a line x = g

is given by

torque =  ( x - g) m( x,y) dA

g =  x

42. Write to Learn: Suppose x is a random variable with probability density p₁( x) , and suppose that y is a random variable with density p₂( y) . If their joint probability density p( x,y) can be written as

p( x,y) = p1( x) p2( y)

then x and y are said to be independent random variables. Write a short essay which shows that if x and y are independent, then

P( a £ x £ b  and  c £ y £ d) = P( a £ x £ b)·P( c £ y £ d)

for all [ a,b] ×[ c,d] .

43. Write to Learn: The convolution for x  > 0 of two functions f(t) and g(t) is given by

(f*g)(t) = t 0  f(t-  u)g(u) du

Show that if X and Y are non-negative random variables with probability densities f and g, respectively, then f*g is the probability density for their sum, X+Y.

44.  What is the probability density for X+Y in exercise 43 if we remove the restriction that X and Y be non-negative.