Information for Problems 1 5 The length x in centimeters of

Information for Problems 1 - 5: The length, x, (in centimeters), of a component manufactured by a particular machine has the following probability distribution function:

Find the length such that 75% of all components have a length greater than that value. (4 points)

Find the mean length. (4 points)

Solution

A)

As by normality conditions,

Integral [f(x)dx] = 1

-C/x | (2,5) = 1

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

C = 10/3

Let a = the critical value.

Thus,

Integral [f(x)dx] | (a, 5) = 0.25

-C/x | (a,5) = 0.25

-C/5 - (-C/a) = 0.25

-(10/3) (1/5 - 1/a) = 0.25

Solving for a,

a = 40/11 [ANSWER]

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Mean = Integral [x f(x) dx]|(2,5)

= C ln(x) |(2,5)

= (10/3) ln(5/2)

= (10/3) ln (2.5) or approx 3.05430244 [answer]