A college professor never finishes his lecture before the end of the hour and al

Question

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 x) is 1.) f(x) f(x) 1.5 1.0 1.0 0.5 0.5 0.5 1.0 1.5 2.0 1.0 1.5 2.0 0.5 f(x) f(x) 1.5 1.0 1.0 0.5 0.5 2.0 0.5 1.0 1.0 1.5 2.0 0.5

Explanation / Answer

a)

Normalizing,

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

= k(2^3)/3 = 1

k(8/3) = 1

k = 3/8=0.375 [ANSWER]

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b)

Here, the cdf is

F(x) = (3/8) x^3 / 3

F(x) = x^3/8

Thus,

P(x<1) = 1^3/8 = 0.125 [ANSWER]

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c)

In minutes, 45 s = 0.75 min, 105 s = 1.5 min. Thus,

P(0.75<x<1.5) = F(1.5) - F(0.45) = 1.5^3/8 - 0.75^3/8

= 0.3691 [ANSWER]

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d)

P(x>1.75) = 1 - F(1.75) = 1 - 1.75^3/8

= 0.3300[ANSWER]

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