The length of time, , needed by students in a particular course to complete a two-hour exam is a random variable with PDF given by:

f(x) = {k(x^2 +x ) 0 ≤ x ≤ 2

{ 0 otherwise

(a) Find the value of k that makes f(x) a valid pdf.

(b) Sketch the pdf.

(c) Find the cumulative distribution function (cdf).

(d) Find the probability that a randomly selected student finishes in less than 1 hour. (Include a sketch and your work.)

(e) Find the probability that a randomly selected student takes between 0.5 and 1.5 hours. (Include a sketch and your work.)

(f) Find the probability that a randomly selected student takes more than 1.5 hours. (Include a sketch and your work.)

(g) Find the expected (mean) amount of time for students to complete the exam. HINT: this question is asking for E(x).

f(x) = {k(x^2 +x ) 0 ≤ x ≤ 2

{ 0 otherwise

(a) Find the value of k that makes f(x) a valid pdf.

(b) Sketch the pdf.

(c) Find the cumulative distribution function (cdf).

(d) Find the probability that a randomly selected student finishes in less than 1 hour. (Include a sketch and your work.)

(e) Find the probability that a randomly selected student takes between 0.5 and 1.5 hours. (Include a sketch and your work.)

(f) Find the probability that a randomly selected student takes more than 1.5 hours. (Include a sketch and your work.)

(g) Find the expected (mean) amount of time for students to complete the exam. HINT: this question is asking for E(x).

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