6. A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea

The three survival functions are e-t, e-2/3t and e-1/3t

Then, the rate of the second death equals (multiplying all combinations of 1 survival + 1 death * the rate of death of the third device)

(e-t*(1-e-2/3t)+(1-e-t)e-2/3t)1/3e-1/3t +

(e-t*(1-e-1/3t)+(1-e-t)e-1/3t)2/3e-2/3t  +

(e-2/3t*(1-e-1/3t)+(1-e-2/3t)e-1/3t)e-t =

1/3e-4/3t + 1/3e-t -2/3e -2t +

2/3e -5/3t + 2/3e-t -4/3e -2t +

e -4/3 + e -5/3 -2e -2t =

e-t + 4/3e-4/3t + 5/3e-5/3t - 4e-2t

Note that this integrates to 1, thus satisfying one of the rules of a density. We also can see that this function is greater than 0, as e-t + 4/3e-4/3t + 5/3e-5/3t - 4e-2t =

e-t + 4/3e-4/3t + 5/3e-5/3t - e-2t - 4/3e-2t - 5/3e-2t =

e-t  - e-2t + 4/3e-4/3t - 4/3e-2t + 5/3e-5/3t- 5/3e-2t =

e-t (1 - e-t) + 4/3e-4/3t(1 - e-2/3t) + 5/3e-5/3t(1 - e-1/3t) > 0 for t > 0

Then, to find the average length that the ship is at sea, we must integrate

t(e-t + 4/3e-4/3t + 5/3e-5/3t - 4e-2t) from 0 to infinity.

However, as a shortcut, note that we can easily identify the integral of the first three functions,

te-t, 4/3te-4/3t, and 5/3te-5/3t, as calculating the expected value of the exponential distribution with parameters 1, 4/3, and 5/3, and the expected value is 1, 3/4, and 3/5

For the final function, 4te-2t = 2*2e-2t , this is twice the expected value of the exponential distribution with parameter 2 = 2 * 1/2 = 1

1 + 3/4 + 3/5 - 1 = 27/20

Step-by-step explanation:

first, we define the variables:

x: number of years after 1950

f (x): amount of vinyl sold.

then, with the variables defined, we have:

68594 vinyl records were sold in 1958 > f (8) = 68594

91299 vinyl records were sold in 1961 > f (11) = 91299

38720 vinyl records were sold in 1952 > f (2) = 38720

161743 vinyl records were sold in 1967 > f (17) = 161743

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step-by-step explanation:

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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