Use a symbolic integration utility to find the mean,...

Use a symbolic integration utility to find the mean, standard deviation, and given probability.
$$
\text { Function } \quad \text { Probability }
$$
$$
f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \quad P(-1.21 \leq x \leq 1.21)
$$

Key Concepts

Normal Distribution

The normal distribution, often called the Gaussian distribution, is a continuous probability distribution with a characteristic bell-shaped curve. It is defined by its mean and standard deviation and is central to probability theory and statistics due to the central limit theorem, which explains why many phenomena tend to follow this distribution.

Symbolic Integration

Symbolic integration refers to the process of finding an exact antiderivative of a function using analytical methods rather than numerical approximations. It is widely used in mathematical analysis and probability theory for obtaining closed-form solutions of integrals, such as those needed to compute probabilities and other statistical measures.

Mean (Expected Value)

The mean or expected value of a random variable is a measure of central tendency that represents the average outcome of an experiment if it were repeated many times. It is calculated by integrating the product of the variable and its probability density function over the entire range of values.

Standard Deviation

The standard deviation is a statistic that quantifies the amount of dispersion or variability in a set of data or a probability distribution. It is calculated as the square root of the variance and provides insight into how spread out the values are around the mean.

Probability Calculation via Integration

Calculating probability in continuous distributions involves integrating the probability density function over a specific interval. This integration process yields the probability that a random variable falls within that interval, making it a fundamental concept in probability theory and statistics.

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