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Statistics / Fundamentals of Probability, with Stochastic Processes 3 / Chapter 6.1 / Problem 2

Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani

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QUESTION:

The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) = 1 16 x2 x 4 0 x < 4. (a) Calculate f , the probability density function of the soap opera. (b) Sketch the graphs of F and f . (c) What is the probability that the soap opera takes at most 50 hours? At least 60 hours? Between 50 and 70 hours? Between 10 and 35 hours?

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QUESTION:

The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) = 1 16 x2 x 4 0 x < 4. (a) Calculate f , the probability density function of the soap opera. (b) Sketch the graphs of F and f . (c) What is the probability that the soap opera takes at most 50 hours? At least 60 hours? Between 50 and 70 hours? Between 10 and 35 hours?

ANSWER:

Step 1 of 3

a.

The distribution function for the duration of a certain soap opera (in tens of hours) is

The probability density function can be obtained by differentiating the cumulative density function .

On differentiating w.r.t x,

Hence, the probability density function is

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