To calculate the probability density function:
\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]
A Probability Density, often referred to in the context of Probability Density Function (PDF), is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete random variable. It describes the likelihood of a random variable taking on a specific value. The area under the curve of a PDF (between any two points) represents the probability that the variable falls within that range. The total area under the curve of a PDF is always equal to 1, representing the total probability of all possible outcomes.
Let's assume the following values:
Step 1: Calculate the exponent part:
\[ -\frac{(x - \mu)^2}{2\sigma^2} = -\frac{(1 - 0)^2}{2 \times 1^2} = -\frac{1}{2} = -0.5 \]
Step 2: Calculate \( e \) raised to the power of the result:
\[ e^{-0.5} \approx 0.60653 \]
Step 3: Multiply by the reciprocal of \( \sigma \sqrt{2\pi} \):
\[ f(x) = \frac{1}{1 \sqrt{2\pi}} \times 0.60653 \approx 0.24197 \]