A normal random variable is a continuous random variable that follows a normal distribution. This means that the data is symmetrically distributed around the mean, with most of the data falling within one standard deviation of the mean. It is often used to model natural phenomena such as human height or IQ scores.
Unlike other types of random variables, a normal random variable has a specific probability distribution known as the normal distribution. This distribution has a bell-shaped curve and is characterized by its mean and standard deviation. Other types of random variables may have different distributions, such as the binomial distribution or the Poisson distribution.
The formula for calculating probabilities for a normal random variable is given by the standard normal distribution formula: P(x < b) = Φ(b), where Φ(b) is the cumulative probability up to the value b. This formula can be used to calculate the probability of a value falling within a certain range, or the probability of a value being greater than or less than a specific value.
A normal random variable is commonly used in statistical analysis to model and analyze real-world data. It allows us to make predictions and draw conclusions about a population based on a sample of data. It is also used in hypothesis testing, where we compare a sample mean to a population mean to determine if there is a significant difference between the two.
Some real-life examples of a normal random variable include human height, IQ scores, and body temperature. These variables are typically normally distributed, meaning that most individuals fall within a certain range of values around the mean. Other examples may include stock market returns, weather patterns, and test scores.