Gaussian Random Variables Question

In summary, a Gaussian random variable is a type of continuous probability distribution that is characterized by a bell-shaped curve and is commonly used to model natural phenomena. It differs from other random variables in that it follows a specific mathematical equation, resulting in a symmetrical curve. The characteristics of a Gaussian random variable include its mean, standard deviation, and shape. The central limit theorem states that many real-world phenomena can be approximated by a Gaussian random variable, making it useful in practical applications such as finance, engineering, and social sciences.
  • #1
chingkui
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How do you show that a linear combination of two Gaussian Random Variables is again Gaussian?
 
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  • #2
By working out what the actual distribution of the linear combination, I'd presume. This homework?
 
  • #3


To show that a linear combination of two Gaussian random variables is again Gaussian, we can use the fact that the sum of two independent Gaussian random variables is also Gaussian.

Let X and Y be two independent Gaussian random variables with means μX and μY, and variances σX^2 and σY^2 respectively.

Now, let Z = aX + bY, where a and b are constants.

We can find the mean and variance of Z as follows:

E[Z] = E[aX + bY] = aE[X] + bE[Y] = aμX + bμY

Var(Z) = Var(aX + bY) = a^2Var(X) + b^2Var(Y) = a^2σX^2 + b^2σY^2

Since X and Y are independent, their joint distribution is also Gaussian. Therefore, the joint probability density function of X and Y can be written as:

fXY(x,y) = (1/2πσXσY)exp[-((x-μX)^2/2σX^2) - ((y-μY)^2/2σY^2)]

Now, using the change of variables method, we can find the probability density function of Z as follows:

fZ(z) = ∫∫fXY(x,y)dxdy

= ∫∫(1/2πσXσY)exp[-((x-μX)^2/2σX^2) - ((y-μY)^2/2σY^2)]dxdy

= (1/2πσXσY)exp[-((z-aμX)^2/2a^2σX^2) - ((z-bμY)^2/2b^2σY^2)]

This is the probability density function of a Gaussian random variable with mean aμX + bμY and variance a^2σX^2 + b^2σY^2, which is the same as the mean and variance we found for Z earlier.

Therefore, Z is a Gaussian random variable and a linear combination of two Gaussian random variables is again Gaussian.
 

1. What is a Gaussian random variable?

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.

2. How is a Gaussian random variable different from other types of random variables?

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.

3. What are the characteristics of a Gaussian random variable?

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.

4. What is the central limit theorem and how does it relate to Gaussian random variables?

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.

5. How are Gaussian random variables used in practical applications?

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.

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