Write the PDF of standard normal distribution calculate the mean and variance also 

The probability density function (PDF) of the standard normal distribution, which is a normal distribution with a mean of 0 and a variance of 1, is given by:

\[ f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \]

### Mean and Variance of Standard Normal Distribution

For the standard normal distribution:

1. **Mean**:
   The mean (μ) of the standard normal distribution is 0.
   \[
   \mu = 0
   \]

2. **Variance**:
   The variance (σ²) of the standard normal distribution is 1.
   \[
   \sigma^2 = 1
   \]

To summarize:
- The PDF of the standard normal distribution is:
  \[
  f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}
  \]

- The mean (\( \mu \)) of the standard normal distribution is 0.
- The variance (\( \sigma^2 \)) of the standard normal distribution is 1.

Would you like further details or have any questions?

Here are some additional questions you might consider:
1. How do you derive the PDF of the normal distribution?
2. What is the cumulative distribution function (CDF) of the standard normal distribution?
3. How can we use the standard normal distribution to find probabilities for a normal distribution with different mean and variance?
4. What are the properties of the normal distribution?
5. How do we standardize a normal variable to convert it to a standard normal variable?

**Tip:** The standard normal distribution is a fundamental concept in statistics, often used for calculating probabilities and z-scores. Understanding its properties can help in various statistical analyses.

Explore the PDF of the standard normal distribution, along with calculations for mean and variance. Understand the formula and properties of the standard normal distribution essential in probability theory and statistics.

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