If f(x)𝑓(𝑥) is a probability density function of a continuous random variable, then ∫∞−∞f(x)dx=∫−∞∞𝑓(𝑥)𝑑𝑥=  ?a.1b.undefinedc.Insufficient datad.0

Break Down the Problem

1. We need to understand the properties of a probability density function (PDF) for a continuous random variable.
2. We are asked to evaluate the integral of the PDF over the entire real line, from $-\infty$ to $+\infty$.

Relevant Concepts

1. A probability density function $f(x)$ must satisfy two properties:
   • $f(x) \geq 0$ for all $x$ (non-negativity).
   • The integral of $f(x)$ over the entire space must equal 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$

Analysis and Detail

1. Since $f(x)$ is a probability density function, by definition, integrating it over all possible values of $x$ gives us the total probability of the distribution.
2. Thus, $\int_{-\infty}^{\infty} f(x) \, dx = 1$

Verify and Summarize

1. The calculation confirms that the integral of a PDF over the entire space is always 1, ensuring the proper normalization condition for probabilities.

Final Answer

The result of the integral is: $\int_{-\infty}^{\infty} f(x) \, dx = 1$ Thus, the answer is a. 1.