A college professor never finishes his lecture before the en

A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is as follows 0 otherwise (a) Find the value of k. (Enter your answer to three decimal places.) Draw the corresponding density curve. [Hint: Total area under the graph of f(x) is 1.] f(x) f(x) WebAssign Plot 1.5 0.5 0.5 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 f(x) f(x) 1.5 0.5 0.5 0.5 2.0 0.5 2.0

Solution

a)

Normalizing,

Integral [f(x)dx] |(0,2) = kx^3/3 |(0,2)

= k(2^3)/3 = 1

k = 3/8 or 0.375 [ANSWER]

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Here, f(2) = 0.75. Thus, it is the UPPER LEFT GRAPH.

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b)

Thus,

F(x) = kx^3/3 = 0.125x^3

Thus,

P(x<1) = F(1) = 0.125 [ANSWER]

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c)

P(0.25<x<0.5) = F(0.5) - F(0.25)

= 0.125*0.5^3 + 0.125*0.25^3

= 0.017578125 [ANSWER]

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d)

P(x>1.5) = 1 - F(1.5)

= 1 - 0.125*1.5^3

= 0.578125 [ANSWER]