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]

2.

3.\displaystyle f(x) = \frac{x}{2} \text{ on } [0, 1]

4.

5.\displaystyle f(x) = \frac{3(x^2-1)}{2} \text{ on } [0,2]

6.

7.\displaystyle f(x) = \frac{1}{x} \text{ on } (0,e]

8.

9.\displaystyle f(x) = e^x \text{ on } [\ln 3,\ln 4]

10.

11.\displaystyle f(x) = -2xe^{-x^2} \text{ on } (-\infty, 0]

12.

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 functionF 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.)

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.