Question

Consider a random continuous variable \(x\) and its cumulative distribution function \(F(x)\). Which of the following options describes the Probability Density Function (PDF) of \(x\)?

• A. \(\dfrac{d}{dx}F(x)\)
• B. \(\dfrac{d^2}{dx^2}F(x)\)
• C. \(\displaystyle\int F(x)\,dx\)
• D. \(\displaystyle\int \dfrac{1}{F(x)}\,dx\)

Hint:

The PDF is the derivative of the CDF with respect to x, by the Fundamental Theorem of Calculus.

Answer 1

The Correct Option is A

Step 1: Recall the definition of the cumulative distribution function (CDF):
For a continuous random variable \(x\), the CDF is defined as \(F(x) = P(X \le x)\), which gives the probability that the random variable takes a value less than or equal to \(x\).
Step 2: Recall the relationship between the CDF and the PDF:
The probability density function \(f(x)\) is defined such that:\[ F(x) = \int_{-\infty}^{x} f(t)\, dt \]
Step 3: Differentiate both sides with respect to \(x\):
By the Fundamental Theorem of Calculus, differentiating the integral with respect to its upper limit gives:\[ \frac{d}{dx}F(x) = f(x) \]
Step 4: Interpret the result:
This shows that the PDF is obtained by taking the first derivative of the CDF, not the second derivative, and not by integrating the CDF.
Step 5: Match with the given options:
The correct expression for the PDF is \(\dfrac{d}{dx}F(x)\), which corresponds to option (A).
Final Answer:
\[ \boxed{f(x) = \frac{d}{dx}F(x)} \]