Question

Let find the following: (a) such that is a probability density function:

a) ∫f(x) dx = 0
b) ∫f(x) dx = 1
c) f(x) ≥ 0 for all x
d) f(x) is continuous for all x

Answer 1

A probability density function (PDF) must satisfy certain conditions, including being integrated to zero over its entire range, integrating to one, being non-negative for all values, and being continuous.

Step-by-step explanation:

A probability density function (PDF) is a function that describes the likelihood of a random variable taking on particular values. To be a valid PDF:

1. a) ∫f(x) dx = 0: The integral of the PDF over its entire range should be equal to zero. This means that there is no probability associated with any single point.
2. b) ∫f(x) dx = 1: The integral of the PDF over its entire range should be equal to one. This ensures that the probabilities of all possible outcomes collectively add up to one.
3. c) f(x) ≥ 0 for all x: The PDF must be non-negative for all possible values of the random variable. Negative probabilities are not meaningful.
4. d) f(x) is continuous for all x: The PDF should be a continuous function, meaning that there are no jumps or discontinuities in its graph.