Question

Which of the following is a probability density function?

1) f(x) = 1/x², x > 1
2) f(x) = 2x, 0 ≤ x ≤ 1
3) f(x) = e⁽⁻ˣ⁾, x ≥ 0
4) f(x) = x², -1 ≤ x ≤ 1

Answer 1

Option 2 (f(x) = 2x, 0 ≤ x ≤ 1) and Option 3 (f(x) = e⁽⁻ˣ⁾, x ≥ 0) are probability density functions.

Step-by-step explanation:

The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a ≤ x ≤ b).

Let's go through the options:

2. f(x) = 1/x², x > 1:
3. This function does not satisfy the condition that f(x) > 0 for all x. Therefore, it is not a probability density function.

f(x) = 2x, 0 ≤ x ≤ 1: This function satisfies the conditions for a probability density function. The area under the curve between 0 and 1 is equal to 1, and f(x) is positive for all x in the given range.f(x) = e⁽⁻ˣ⁾, x ≥ 0: This function satisfies the conditions for a probability density function. The area under the curve is equal to 1, and f(x) is positive for all x ≥ 0.f(x) = x², -1 ≤ x ≤ 1: This function does not satisfy the condition that f(x) > 0 for all x. Therefore, it is not a probability density function.

Based on the conditions for a probability density function, options 2 and 3 are probability density functions.