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