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solve this in detail with explanation




To address the problem, we need to write the probability density function (pdf) of the standard normal distribution and then calculate its mean and variance. Let's break it down step by step.

### Step 1: Write the PDF of the Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. The probability density function (pdf) of the standard normal distribution is given by:

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

Where:
- \( x \) is the variable.
- \( \sqrt{2\pi} \) is the square root of \(2\pi\) which normalizes the distribution.

### Step 2: Calculate the Mean of the Standard Normal Distribution

The mean (\(\mu\)) of a standard normal distribution is known to be 0. We can also confirm this mathematically by integrating the product of \(x\) and the pdf over the entire range of \(x\) (from \(-\infty\) to \(\infty\)):

\[
\mu = \int_{-\infty}^{\infty} x f(x) dx = \int_{-\infty}^{\infty} x \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx
\]

Since the standard normal distribution is symmetric around 0, the mean, \( \mu \), is 0.

### Step 3: Calculate the Variance of the Standard Normal Distribution

The variance (\(\sigma^2\)) of the standard normal distribution is 1. This can also be calculated by integrating the square of the difference between \(x\) and the mean, weighted by the pdf, over the entire range of \(x\):

\[
\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx
\]

Since \(\mu = 0\), this simplifies to:

\[
\sigma^2 = \int_{-\infty}^{\infty} x^2 \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx
\]

Through a standard calculation (often using integration by parts or recognizing it as a known Gaussian integral), this integral evaluates to 1.

### Summary
- **PDF of Standard Normal Distribution**: \( f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \)
- **Mean (\(\mu\))**: 0
- **Variance (\(\sigma^2\))**: 1

Would you like more details or have any questions?

### Related Questions:
1. What is the cumulative distribution function (CDF) of the standard normal distribution?
2. How does the standard normal distribution relate to the general normal distribution?
3. What is the moment generating function of the standard normal distribution?
4. How do you convert a normal random variable to a standard normal variable?
5. What is the skewness of the standard normal distribution?
6. How is the standard normal distribution used in hypothesis testing?
7. Can the standard normal distribution be used to approximate other distributions?
8. How does the area under the standard normal curve relate to probability?

### Tip:
When dealing with the normal distribution, always remember that for the standard normal distribution, the tables often used (Z-tables) already assume a mean of 0 and a variance of 1, which simplifies many calculations.

Explore the probability density function (PDF), mean, and variance of the standard normal distribution. Learn how these fundamental concepts are derived and apply to statistical analysis. Suitable for advanced high school students and beyond.

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