Questions: Which of the following are properties of a probability density function (pdf)? Select all that apply. A. The probability that x takes on any single individual value is greater than 0. B. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable. C. The high point of the graph must be at the value of the population standard deviation, σ. D. The total area under the graph of the equation over all possible values of the random variable must equal 1. E. The values of the random variable must be greater than or equal to 0. F. The graph of the probability density function must be symmetric.

Which of the following are properties of a probability density function (pdf)?

Select all that apply.
A. The probability that x takes on any single individual value is greater than 0.
B. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable.
C. The high point of the graph must be at the value of the population standard deviation, σ.
D. The total area under the graph of the equation over all possible values of the random variable must equal 1.
E. The values of the random variable must be greater than or equal to 0.
F. The graph of the probability density function must be symmetric.

Transcript text: Which of the following are properties of a probability density function (pdf)? Select all that apply. A. The probability that $x$ takes on any single individual value is greater than 0. B. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable. C. The high point of the graph must be at the value of the population standard deviation, $\sigma$. D. The total area under the graph of the equation over all possible values of the random variable must equal 1. E. The values of the random variable must be greater than or equal to 0. F. The graph of the probability density function must be symmetric.

Solution

Solution Steps

Step 1: Verify the total area under the graph of the function over all possible values equals 1.

This ensures that the total probability space is accounted for. The formula to check this is the integral of the pdf over its entire range should equal 1.

Step 2: Ensure the height of the graph of the function is greater than or equal to 0 for all possible values.

This is because probabilities cannot be negative. This is verified by checking the function's value at various points in its domain.

Step 3: Confirm that the probability of the random variable taking on any single individual value is 0.

This is a fundamental property of continuous random variables, which are described by pdfs. It's verified based on the definition of a pdf.

Final Answer:

If all the above conditions are met, the function can be considered a valid probability density function.

Step 1: Total Area Under the Graph

The total area under the graph is 1, which satisfies the first property of a valid pdf.

Step 2: Height of the Graph

The height of the graph is non-negative for all values, satisfying the second property of a valid pdf.

Step 3: Probability of a Single Value

The probability of the random variable taking on any single individual value is 0, satisfying the third property of a valid pdf.

Final Answer:

Based on the provided parameters, the function satisfies the necessary properties to be considered a valid probability density function (pdf).

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