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: Start from the limit definition of the CDF slope:
By definition, $F(x) = P(X \le x)$. Consider a small interval of width $\Delta x$ around a point $x$.
Step 2: Express the probability of falling in this small interval:
$$P(x < X \le x+\Delta x) = F(x+\Delta x) - F(x)$$
Step 3: Relate this probability to the density function:
For a continuous variable, this probability is approximately $f(x)\Delta x$ for small $\Delta x$, since the density $f(x)$ represents probability per unit length:$$f(x)\Delta x \approx F(x+\Delta x) - F(x)$$
Step 4: Divide both sides by $\Delta x$ and take the limit as $\Delta x \to 0$:
$$f(x) = \lim_{\Delta x \to 0} \frac{F(x+\Delta x)-F(x)}{\Delta x}$$This limit is exactly the definition of the derivative of $F(x)$ with respect to $x$.
Step 5: Conclude the relationship:
$$f(x) = \frac{d}{dx}F(x)$$This confirms that the PDF is the first derivative of the CDF, so option (A) is correct.
Final Answer:
\[ \boxed{f(x) = \frac{d}{dx}F(x)} \]