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 Applied calculus exercises: probability density functions and the uniform distribution

In Exercises 1–12, decide whether the given function is a probability density function. If a function fails to be a probability density function, select a reason.

 1. f(x) = 1 \text{ on } [0, 1] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 2. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 3. \displaystyle f(x) = \frac{x}{2} \text{ on } [0, 1] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 4. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 5. \displaystyle f(x) = \frac{3(x^2-1)}{2} \text{ on } [0,2] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 6. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 7. \displaystyle f(x) = \frac{1}{x} \text{ on } (0,e] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 8. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 9. \displaystyle f(x) = e^x \text{ on } [\ln 3,\ln 4] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 10. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 11. \displaystyle f(x) = -2xe^{-x^2} \text{ on } (-\infty, 0] Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways 12. Select one is a probability density function fails only to be nonnegative fails only to have an integral of 1 fails in both ways

In Exercises 13–18, find the values of k for which the given functions are probability density functions.

 13. f(x) = 2k \text{ on } [-1, 1] k = 14. k = 15. f(x) = ke^{kx} \text{ on } [0, 1] k = 16. k = 17. \displaystyle f(x) = \frac{k}{x^{3/2}} \text{ on } [1, +\infty) k = 18. k =

Cumulative Distribution If f is a probability density function defined on the interval (a, b), then the associated cumulative distribution function F is given by

F(x) = \int_a^x f(t)\ dt
 \displaystyle F(x) = \text{ Shaded Area } =\int_0^xf(t)\ dt

19. Why is F'(x) = f(x) ?

20. Use the result of the previous exercise to show that

P(c \leq x \leq d) = F(d) - F(c)

for a \leq c \leq d \leq b.

21. Show that F(a) = 0 and F(b) = 1.

22. Can F(x) can have any relative extrema in the interior of its domain? (Give a reason for your answer.)

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Applications

Unless otherwise stated, round answers to all applications to four decimal places.

T Shirts The age (in years) of randomly chosen T shirts in your wardrobe from last summer is distributed according to the density function \displaystyle f(x) = \frac{10}{9x^2} with 1 \leq x \leq 10. Find the probability that a randomly chosen T-shirt is between 2 and 8 years old.

Entertainment The number of hours you spend looking at YouTube on a typical Saturday night is distributed according to the density function f(x) = 2xe^{-x^2} with 0 \leq x. Find the probability that, on a typical Saturday night, you spend between and hours watching YouTube.

The Doomsday Meteor The probability that a "doomsday meteor" will hit the earth in any given year and release a billion megatons or more of energy is on the order of 0.000 000 01. If X is the year in which a doomsday meteor hits the earth, then is may be modeled with an associated probability density function given by f(x) = ae^{-ax} with a = 0.000\ 000\ 01 (see the discussion of exponential probability density functions in the next section to see where this density function comes from).

(a) What is the probability that the earth will be hit by a doomsday meteor at least once during the next 100 years? (Give the answer correct to 2 significant digits.)

(b) What is the probability that the earth has been hit by a doomsday meteor at least once since the appearance of life (about 4 billion years ago)?

Source: NASA International Near-Earth-Object Detection Workshop (The New York Times, January 25, 1994, p. C1.)

Galactic Cataclysm The probability that the galaxy MX-47 will explode within the next million years is estimated to be . If X is the million-year period in which the galaxy MX-47 will explode, then is may be modeled with an associated probability density function given by f(x) = ae^{-ax} with a = (see the discussion of exponential probability density functions in the next section to see where this density function comes from).

(a) What is the probability that MX-47 will explode within the next million years? (Give the answer correct to two significant digits.)

(b) What is the probability that MX-47 will still be around million years hence? (Give the answer correct to two significant digits.)

Salaries Assuming that workers' salaries in your company are uniformly distributed between $10,000 and$40,000 per year, find the probability that a randomly chosen worker earns an annual salary between $14,000 and$20,000.

Grades The grade point averages of members of the Gourmet Society are uniformly distributed between and . Find the probability that a randomly chosen member of the society has a grade point average between and .

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Communication and Reasoning Exercises

Why is a probability density function often more convenient than a histogram?

Give an example of a probability density function that is increasing everywhere on its domain.

Give an example of a probability density function that is concave up everywhere on its domain.

Give an example of a continuous random variable that does not admit a probability density function. [Hint: See the note in the text after the definition of a probability density function.]

Your friend thinks that if f is a probability density function for the continuous random variable X, then for each value a, f(a) is the probability that X = a. Explain to your friend why this is wrong.

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Last Updated: March, 2008