1. **Problem Statement:** We will learn about the Normal Distribution, a key concept in probability and statistics used to calculate probabilities for continuous random variables. 2. **Concept and Definition:** The Normal Distribution, also called Gaussian distribution, is a continuous probability distribution characterized by a symmetric bell-shaped curve. It is defined by two parameters: the mean $\mu$ (center of the distribution) and the standard deviation $\sigma$ (spread or dispersion). 3. **Formula:** The probability density function (pdf) of a normal distribution is given by: $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ This function describes the likelihood of the random variable $X$ taking a value near $x$. 4. **Standard Normal Distribution:** When $\mu=0$ and $\sigma=1$, the distribution is called the standard normal distribution, denoted by $Z$. Probabilities for any normal distribution can be found by converting to $Z$ using the formula: $$Z = \frac{X - \mu}{\sigma}$$ 5. **Calculating Probabilities:** To find the probability that $X$ lies between two values $a$ and $b$, calculate: $$P(a \leq X \leq b) = P\left(\frac{a-\mu}{\sigma} \leq Z \leq \frac{b-\mu}{\sigma}\right)$$ Use standard normal distribution tables or software to find these probabilities. 6. **Procedure:** - Identify $\mu$ and $\sigma$ of the distribution. - Convert the values $a$ and $b$ to their corresponding $Z$ scores. - Use the standard normal table or calculator to find probabilities for these $Z$ values. - Subtract to find the probability between $a$ and $b$. 7. **Important Rules:** - Total area under the normal curve is 1. - The curve is symmetric about $\mu$. - Approximately 68% of data lies within $\mu \pm \sigma$, 95% within $\mu \pm 2\sigma$, and 99.7% within $\mu \pm 3\sigma$. This explanation prepares you to solve problems involving normal distribution probabilities. Please provide sums to proceed with examples and solutions.