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How do you show that a linear combination of two Gaussian Random Variables is again Gaussian?
A Gaussian random variable is a type of continuous probability distribution that is commonly used in statistics and data analysis. It is also known as a normal distribution and is characterized by a bell-shaped curve. It is often used to model natural phenomena such as height, weight, and test scores.
A Gaussian random variable is different from other types of random variables in that it follows a specific mathematical equation that results in a symmetrical, bell-shaped curve. This means that the majority of the values will be clustered around the mean, with fewer values in the tails of the distribution.
A Gaussian random variable is characterized by its mean, standard deviation, and shape. The mean represents the center of the distribution, while the standard deviation measures the spread of the data. The shape of the curve is always symmetrical, with the mean in the center and the tails of the curve approaching but never touching the x-axis.
The central limit theorem states that when a large enough sample size is taken from any population, the sampling distribution of the sample means will approximate a normal distribution, regardless of the shape of the original population. This means that many real-world phenomena can be approximated by a Gaussian random variable.
Gaussian random variables have a wide range of practical applications in various fields such as finance, engineering, and social sciences. They are used to model real-world phenomena and make predictions about future outcomes. They are also used in statistical tests and analyses to make inferences about a population based on a sample of data.