Question

For a continuous probability distribution, what is the minimum value of the probability density function?

1) 0
2) 1
3) Cannot be determined
4) Infinity

Answer 1

The minimum value of a probability density function for a continuous distribution is 0. For a uniform distribution defined from 0 to 10, the probability of the interval between 0 and 4 is 0.4.

Step-by-step explanation:

The minimum value of a probability density function (PDF) for a continuous probability distribution is 0. This is because the PDF represents how likely it is for a value in the distribution to occur within a small interval, hence its minimum value can't be negative as it would imply negative likelihood which is not possible in probability theory. For example, the probability P(0 < x < 4) for a uniform continuous probability distribution where f(x) = 1 for 0 ≤ x ≤ 10, would be calculated as the area under f(x) between x = 0 and x = 4. This corresponds to 4 units of base times the height of 1 unit, resulting in an area, or probability, of 0.4.

Answering the specific questions given:

1. For the continuous probability distribution, the PDF cannot take a value lower than 0, hence option 1) 0 is correct.
2. The probability P(x > 15) for a distribution defined only up to x = 15 is 0.
3. The area under the probability density function for a continuous probability distribution is always equal to 1, representing the total probability.
4. For a continuous distribution, P(x = 7), or any specific value, is always 0 because a PDF defines probabilities for intervals, not for specific points.
5. If a continuous probability function is restricted to x = 0 and x = 7, then P(x = 10) is also 0, as 10 is not within the described interval.
6. For the continuous function restricted to 0 ≤ x ≤ 5, P(x < 0) is 0, since no probability mass lies below zero.