18.1 Probability Density Function (PDF) of the Normal Distribution

The Probability Density Function (PDF) for a normal distribution quantifies the likelihood that a continuous random variable will assume a particular value. This function graphically generates the iconic bell-shaped curve, which is perfectly symmetrical around the distribution's mean ($\mu$). The standard deviation ($\sigma$) plays a crucial role in determining the spread or width of this curve.

PDF Formula for the Normal Distribution

The mathematical formula for the PDF of a normal distribution is:

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$$

Where:

• $f(x)$: The probability density at a specific value $x$.
• $\mu$ (mu): The mean of the distribution. This represents the central tendency or average value of the data.
• $\sigma$ (sigma): The standard deviation of the distribution. This measures the amount of variation or dispersion of the data points from the mean.
• $\pi$ (pi): The mathematical constant, approximately 3.14159.
• $e$: Euler's number, the base of the natural logarithm, approximately 2.71828.
• $x$: The value of the random variable, which can be any real number (ranging from $-\infty$ to $+\infty$).

Key Characteristics of the Normal Distribution PDF

• Peak at the Mean: The curve reaches its highest point at $x = \mu$, indicating that the mean is the most probable value for the random variable.
• Symmetry: The distribution is perfectly symmetrical around the mean ($\mu$). This means that the probability of observing a value $x$ units above the mean is the same as the probability of observing a value $x$ units below the mean.
• Spread Determined by Standard Deviation:
  • A smaller $\sigma$ results in a narrower and taller curve, indicating that data points are clustered closely around the mean.
  • A larger $\sigma$ results in a wider and flatter curve, signifying that data points are more spread out from the mean.
• Total Area Under the Curve: The total area under the PDF curve for any probability distribution is always equal to 1. This represents the certainty that the random variable will take on some value within its possible range.

Understanding the PDF Value

It's important to note that for a continuous distribution like the normal distribution, the probability of the random variable taking on any exact single value is technically zero. Instead, the PDF value $f(x)$ represents the probability density at point $x$. The probability of the random variable falling within a specific range $[a, b]$ is found by integrating the PDF from $a$ to $b$: $P(a \le X \le b) = \int_{a}^{b} f(x) dx$.

Practical Applications

The normal distribution and its PDF are fundamental in statistics and are used to model a wide variety of natural phenomena and data sets, including:

• Heights and weights of populations
• Measurement errors
• Exam scores
• Financial market returns (under certain assumptions)

Common Interview Questions

• What is the formula for the probability density function (PDF) of a normal distribution?
• How do the mean ($\mu$) and standard deviation ($\sigma$) influence the shape and position of the normal distribution curve?
• Why is the normal distribution curve symmetrical around the mean?
• What does the total area under the normal distribution's PDF represent, and why is it significant?
• How does an increase in the standard deviation affect the appearance and interpretation of the normal distribution PDF?
• What is the significance of the peak of the normal distribution curve?
• How do you interpret the value of the PDF at a specific point $x$? What does $f(x)$ actually signify?
• Why can the values of a normally distributed random variable range from negative infinity to positive infinity?
• What roles do the mathematical constants $\pi$ and $e$ play in the formulation of the normal distribution PDF?
• Can you provide examples of real-world phenomena that are often modeled using the normal distribution?