The length x in centimeters of a component manufactured by a

The length, x, (in centimeters), of a component manufactured by a particular machine has the following probability distribution function: f(x) = C 1/x2 for 2 less-than or equal x less-than or equal 5 Find the value of C that makes this a legitimate probability distribution function. Find the cumulative distribution function F(x). Find the probability that the length of the component is greater than 3 cm. Find the length such that 75% of all components have a length greater than that value. Find the mean length.

Solution

a)

The integral of f(x) must sum up to 1.

Thus,

Integral [f(x) dx] = -C/x |(2,5) = 1

-C/5 - (-C/2) = 1

3C/10 = 1

C = 10/3 [answer]

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b)

At = 2, the cumulative function must be 0.

hence,

-10/3x + D = 0, x = 2

Hence,

D = 5/3

F(x) = 0, x<2
-10/(3x) + 5/3, 2<=x<=5
1, x>5 [ANSWER]

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C)

P(x>3) = 1- F(3) = 1 - [-10/(3*3) + 5/3] = 4/9 [ANSWER]

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d)

Thus, 25% is less than that value.

F(x) = 0.25 = -10/(3x) + 5/3

1/4 = -10/(3x) + 5/3

10/(3x) = 17/12

10/x = 17/4

x = 40/17 [answer]

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e)

Mean = Integral [x f(x) dx] = (10/3) ln|x| |(2,5)

= (10/3) ln (5/2) = 3.05430244 [answer]